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Theorem preimaf1ofi 6807
Description: The preimage of a finite set under a one-to-one, onto function is finite. (Contributed by Jim Kingdon, 24-Sep-2022.)
Hypotheses
Ref Expression
preimaf1ofi.ss  |-  ( ph  ->  C  C_  B )
preimaf1ofi.f  |-  ( ph  ->  F : A -1-1-onto-> B )
preimaf1ofi.c  |-  ( ph  ->  C  e.  Fin )
Assertion
Ref Expression
preimaf1ofi  |-  ( ph  ->  ( `' F " C )  e.  Fin )

Proof of Theorem preimaf1ofi
StepHypRef Expression
1 preimaf1ofi.c . 2  |-  ( ph  ->  C  e.  Fin )
2 preimaf1ofi.f . . . 4  |-  ( ph  ->  F : A -1-1-onto-> B )
3 f1ocnv 5348 . . . 4  |-  ( F : A -1-1-onto-> B  ->  `' F : B -1-1-onto-> A )
4 f1of1 5334 . . . 4  |-  ( `' F : B -1-1-onto-> A  ->  `' F : B -1-1-> A
)
52, 3, 43syl 17 . . 3  |-  ( ph  ->  `' F : B -1-1-> A
)
6 preimaf1ofi.ss . . 3  |-  ( ph  ->  C  C_  B )
7 f1ores 5350 . . 3  |-  ( ( `' F : B -1-1-> A  /\  C  C_  B )  ->  ( `' F  |`  C ) : C -1-1-onto-> ( `' F " C ) )
85, 6, 7syl2anc 408 . 2  |-  ( ph  ->  ( `' F  |`  C ) : C -1-1-onto-> ( `' F " C ) )
9 f1ofi 6799 . 2  |-  ( ( C  e.  Fin  /\  ( `' F  |`  C ) : C -1-1-onto-> ( `' F " C ) )  -> 
( `' F " C )  e.  Fin )
101, 8, 9syl2anc 408 1  |-  ( ph  ->  ( `' F " C )  e.  Fin )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1465    C_ wss 3041   `'ccnv 4508    |` cres 4511   "cima 4512   -1-1->wf1 5090   -1-1-onto->wf1o 5092   Fincfn 6602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-er 6397  df-en 6603  df-fin 6605
This theorem is referenced by:  fisumss  11129
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