Theorem f1veqaeq 5944

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 5943	. . 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 5672	. . . . . . . 8 |- ( c = C -> ( F ` c ) = ( F ` C ) )
3	2	eqeq1d 2243	. . . . . . 7 |- ( c = C -> ( ( F ` c ) = ( F ` d ) <-> ( F ` C ) = ( F ` d ) ) )
4		eqeq1 2241	. . . . . . 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 5672	. . . . . . . 8 |- ( d = D -> ( F ` d ) = ( F ` D ) )
7	6	eqeq2d 2246	. . . . . . 7 |- ( d = D -> ( ( F ` C ) = ( F ` d ) <-> ( F ` C ) = ( F ` D ) ) )
8		eqeq2 2244	. . . . . . 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 2936	. . . . 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 1398   e. wcel 2205   A.wral 2522   -->wf 5350   -1-1->wf1 5351   ` cfv 5354

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fv 5362

This theorem is referenced by:  f1fveq  5947  f1ocnvfvrneq  5957  f1o2ndf1  6426  1dom1el  7062  fidceq  7126  difinfsnlem  7392  difinfsn  7393  pr2cv1  7494  iseqf1olemab  10868  iseqf1olemnanb  10869  f1ghm0to0  14006  uspgr2wlkeq  16377  3dom  16779  pwle2  16789