Theorem f1veqaeq 5548

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 5547	. . 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 5305	. . . . . . . 8 |- ( c = C -> ( F ` c ) = ( F ` C ) )
3	2	eqeq1d 2096	. . . . . . 7 |- ( c = C -> ( ( F ` c ) = ( F ` d ) <-> ( F ` C ) = ( F ` d ) ) )
4		eqeq1 2094	. . . . . . 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 5305	. . . . . . . 8 |- ( d = D -> ( F ` d ) = ( F ` D ) )
7	6	eqeq2d 2099	. . . . . . 7 |- ( d = D -> ( ( F ` C ) = ( F ` d ) <-> ( F ` C ) = ( F ` D ) ) )
8		eqeq2 2097	. . . . . . 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 2734	. . . . 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 1289   e. wcel 1438   A.wral 2359   -->wf 5011   -1-1->wf1 5012   ` cfv 5015

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fv 5023

This theorem is referenced by:  f1fveq  5551  f1ocnvfvrneq  5561  f1o2ndf1  5993  fidceq  6585  iseqf1olemab  9918  iseqf1olemnanb  9919