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Theorem tz6.12-1 3721
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27.
Hypothesis
Ref Expression
tz6.12.1 |- A e. V
Assertion
Ref Expression
tz6.12-1 |- ((AFy /\ E!y AFy) -> (F` A) = y)
Distinct variable groups:   y,F   y,A
Proof of Theorem tz6.12-1
StepHypRef Expression
1 tz6.12.1 . . . . . . . 8 |- A e. V
21fv3 3718 . . . . . . 7 |- (F` A) = {z | (E.y(z e. y /\ AFy) /\ E!y AFy)}
32abeq2i 1562 . . . . . 6 |- (z e. (F` A) <-> (E.y(z e. y /\ AFy) /\ E!y AFy))
4 exancom 1050 . . . . . . . . 9 |- (E.y(z e. y /\ AFy) <-> E.y(AFy /\ z e. y))
54anbi1i 480 . . . . . . . 8 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) <-> (E.y(AFy /\ z e. y) /\ E!y AFy))
6 ancom 435 . . . . . . . 8 |- ((E.y(AFy /\ z e. y) /\ E!y AFy) <-> (E!y AFy /\ E.y(AFy /\ z e. y)))
75, 6bitr 173 . . . . . . 7 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) <-> (E!y AFy /\ E.y(AFy /\ z e. y)))
8 eupick 1427 . . . . . . 7 |- ((E!y AFy /\ E.y(AFy /\ z e. y)) -> (AFy -> z e. y))
97, 8sylbi 199 . . . . . 6 |- ((E.y(z e. y /\ AFy) /\ E!y AFy) -> (AFy -> z e. y))
103, 9sylbi 199 . . . . 5 |- (z e. (F` A) -> (AFy -> z e. y))
1110com12 11 . . . 4 |- (AFy -> (z e. (F` A) -> z e. y))
1211adantr 389 . . 3 |- ((AFy /\ E!y AFy) -> (z e. (F` A) -> z e. y))
13 19.8a 1025 . . . . . . 7 |- ((z e. y /\ AFy) -> E.y(z e. y /\ AFy))
1413anim1i 334 . . . . . 6 |- (((z e. y /\ AFy) /\ E!y AFy) -> (E.y(z e. y /\ AFy) /\ E!y AFy))
1514anasss 440 . . . . 5 |- ((z e. y /\ (AFy /\ E!y AFy)) -> (E.y(z e. y /\ AFy) /\ E!y AFy))
1615, 3sylibr 200 . . . 4 |- ((z e. y /\ (AFy /\ E!y AFy)) -> z e. (F` A))
1716expcom 374 . . 3 |- ((AFy /\ E!y AFy) -> (z e. y -> z e. (F` A)))
1812, 17impbid 514 . 2 |- ((AFy /\ E!y AFy) -> (z e. (F` A) <-> z e. y))
1918eqrdv 1466 1 |- ((AFy /\ E!y AFy) -> (F` A) = y)
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977  E!weu 1373  Vcvv 1802   class class class wbr 2609  ` cfv 3172
This theorem is referenced by:  tz6.12 3722  tz6.12c 3725  funbrfv 3735
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-xp 3174  df-cnv 3176  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fv 3188
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