Theorem f1veqaeq 5429

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 5428	. . 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 5198	. . . . . . . 8 |- ( c = C -> ( F ` c ) = ( F ` C ) )
3	2	eqeq1d 2089	. . . . . . 7 |- ( c = C -> ( ( F ` c ) = ( F ` d ) <-> ( F ` C ) = ( F ` d ) ) )
4		eqeq1 2087	. . . . . . 7 |- ( c = C -> ( c = d <-> C = d ) )
5	3, 4	imbi12d 232	. . . . . 6 |- ( c = C -> ( ( ( F ` c ) = ( F ` d ) -> c = d ) <-> ( ( F ` C ) = ( F ` d ) -> C = d ) ) )
6		fveq2 5198	. . . . . . . 8 |- ( d = D -> ( F ` d ) = ( F ` D ) )
7	6	eqeq2d 2092	. . . . . . 7 |- ( d = D -> ( ( F ` C ) = ( F ` d ) <-> ( F ` C ) = ( F ` D ) ) )
8		eqeq2 2090	. . . . . . 7 |- ( d = D -> ( C = d <-> C = D ) )
9	7, 8	imbi12d 232	. . . . . 6 |- ( d = D -> ( ( ( F ` C ) = ( F ` d ) -> C = d ) <-> ( ( F ` C ) = ( F ` D ) -> C = D ) ) )
10	5, 9	rspc2v 2713	. . . . 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 271	. . 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 119	. 2 |- ( F : A -1-1-> B -> ( ( C e. A /\ D e. A ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) ) )
14	13	imp 122	1 |- ( ( F : A -1-1-> B /\ ( C e. A /\ D e. A ) ) -> ( ( F ` C ) = ( F ` D ) -> C = D ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   A.wral 2348   -->wf 4918   -1-1->wf1 4919   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fv 4930

This theorem is referenced by:  f1fveq  5432  f1ocnvfvrneq  5442  f1o2ndf1  5869  fidceq  6354