Theorem f1veqaeq 5748

Description: If the values of a one-to-one function for two arguments are equal, the arguments themselves must be equal. (Contributed by Alexander van der Vekens, 12-Nov-2017.)

Assertion

Ref	Expression
f1veqaeq	|- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) )

Proof of Theorem f1veqaeq

Dummy variables c d are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dff13 5747	. . 3 |- ( F : A -1-1-> B <-> ( F : A --> B /\ A. c e. A A. d e. A ( ( F ` c ) = ( F ` d ) -> c = d ) ) )
2		fveq2 5496	. . . . . . . 8 |- ( c = C -> ( F ` c ) = ( F ` C ) )
3	2	eqeq1d 2179	. . . . . . 7 |- ( c = C -> ( ( F ` c ) = ( F ` d ) <-> ( F ` C ) = ( F ` d ) ) )
4		eqeq1 2177	. . . . . . 7 |- ( c = C -> ( c = d <-> C = d ) )
5	3, 4	imbi12d 233	. . . . . 6 |- ( c = C -> ( ( ( F ` c ) = ( F ` d ) -> c = d ) <-> ( ( F ` C ) = ( F ` d ) -> C = d ) ) )
6		fveq2 5496	. . . . . . . 8 |- ( d = D -> ( F ` d ) = ( F ` D ) )
7	6	eqeq2d 2182	. . . . . . 7 |- ( d = D -> ( ( F ` C ) = ( F ` d ) <-> ( F ` C ) = ( F ` D ) ) )
8		eqeq2 2180	. . . . . . 7 |- ( d = D -> ( C = d <-> C = D ) )
9	7, 8	imbi12d 233	. . . . . 6 |- ( d = D -> ( ( ( F ` C ) = ( F ` d ) -> C = d ) <-> ( ( F ` C ) = ( F ` D ) -> C = D ) ) )
10	5, 9	rspc2v 2847	. . . . 5 |- ( ( C e. A /\ D e. A ) -> ( A. c e. A A. d e. A ( ( F ` c ) = ( F ` d ) -> c = d ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) ) )
11	10	com12 30	. . . 4 |- ( A. c e. A A. d e. A ( ( F ` c ) = ( F ` d ) -> c = d ) -> ( ( C e. A /\ D e. A ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) ) )
12	11	adantl 275	. . 3 |- ( ( F : A --> B /\ A. c e. A A. d e. A ( ( F ` c ) = ( F ` d ) -> c = d ) ) -> ( ( C e. A /\ D e. A ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) ) )
13	1, 12	sylbi 120	. 2 |- ( F : A -1-1-> B -> ( ( C e. A /\ D e. A ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) ) )
14	13	imp 123	1 |- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   A.wral 2448   -->wf 5194   -1-1->wf1 5195   ` cfv 5198

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fv 5206

This theorem is referenced by:  f1fveq  5751  f1ocnvfvrneq  5761  f1o2ndf1  6207  fidceq  6847  difinfsnlem  7076  difinfsn  7077  iseqf1olemab  10445  iseqf1olemnanb  10446  pwle2  14031