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Theorem fvimacnvi 3789
Description: A member of a preimage is a function value argument.
Assertion
Ref Expression
fvimacnvi |- ((Fun F /\ A e. (`'F"B)) -> (F` A) e. B)
Proof of Theorem fvimacnvi
StepHypRef Expression
1 funimass2 3559 . . 3 |- ((Fun F /\ {A} (_ (`'F"B)) -> (F"{A}) (_ B)
2 snssi 2457 . . 3 |- (A e. (`'F"B) -> {A} (_ (`'F"B))
31, 2sylan2 451 . 2 |- ((Fun F /\ A e. (`'F"B)) -> (F"{A}) (_ B)
4 fnsnfv 3752 . . . . . 6 |- ((F Fn dom F /\ A e. dom F) -> {(F` A)} = (F"{A}))
5 funfn 3528 . . . . . 6 |- (Fun F <-> F Fn dom F)
64, 5sylanb 449 . . . . 5 |- ((Fun F /\ A e. dom F) -> {(F` A)} = (F"{A}))
7 cnvimass 3407 . . . . . 6 |- (`'F"B) (_ dom F
87sseli 2055 . . . . 5 |- (A e. (`'F"B) -> A e. dom F)
96, 8sylan2 451 . . . 4 |- ((Fun F /\ A e. (`'F"B)) -> {(F` A)} = (F"{A}))
109sseq1d 2078 . . 3 |- ((Fun F /\ A e. (`'F"B)) -> ({(F` A)} (_ B <-> (F"{A}) (_ B))
11 fvex 3717 . . . 4 |- (F` A) e. V
1211snss 2452 . . 3 |- ((F` A) e. B <-> {(F` A)} (_ B)
1310, 12syl5bb 530 . 2 |- ((Fun F /\ A e. (`'F"B)) -> ((F` A) e. B <-> (F"{A}) (_ B))
143, 13mpbird 196 1 |- ((Fun F /\ A e. (`'F"B)) -> (F` A) e. B)
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955   (_ wss 2037  {csn 2399  `'ccnv 3159  dom cdm 3160  "cima 3163  Fun wfun 3166   Fn wfn 3167  ` cfv 3172
This theorem is referenced by:  fvimacnv 3790
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  ax-un 2857
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-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-fv 3188
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