Theorem f1veqaeq 5770

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 5769	. . 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 5516	. . . . . . . 8 |- ( c = C -> ( F ` c ) = ( F ` C ) )
3	2	eqeq1d 2186	. . . . . . 7 |- ( c = C -> ( ( F ` c ) = ( F ` d ) <-> ( F ` C ) = ( F ` d ) ) )
4		eqeq1 2184	. . . . . . 7 |- ( c = C -> ( c = d <-> C = d ) )
5	3, 4	imbi12d 234	. . . . . 6 |- ( c = C -> ( ( ( F ` c ) = ( F ` d ) -> c = d ) <-> ( ( F ` C ) = ( F ` d ) -> C = d ) ) )
6		fveq2 5516	. . . . . . . 8 |- ( d = D -> ( F ` d ) = ( F ` D ) )
7	6	eqeq2d 2189	. . . . . . 7 |- ( d = D -> ( ( F ` C ) = ( F ` d ) <-> ( F ` C ) = ( F ` D ) ) )
8		eqeq2 2187	. . . . . . 7 |- ( d = D -> ( C = d <-> C = D ) )
9	7, 8	imbi12d 234	. . . . . 6 |- ( d = D -> ( ( ( F ` C ) = ( F ` d ) -> C = d ) <-> ( ( F ` C ) = ( F ` D ) -> C = D ) ) )
10	5, 9	rspc2v 2855	. . . . 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 277	. . 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 121	. 2 |- ( F : A -1-1-> B -> ( ( C e. A /\ D e. A ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) ) )
14	13	imp 124	1 |- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   -->wf 5213   -1-1->wf1 5214   ` cfv 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fv 5225

This theorem is referenced by:  f1fveq  5773  f1ocnvfvrneq  5783  f1o2ndf1  6229  fidceq  6869  difinfsnlem  7098  difinfsn  7099  iseqf1olemab  10489  iseqf1olemnanb  10490  pwle2  14751