Theorem f1imaeq 5743

Description: Taking images under a one-to-one function preserves equality. (Contributed by Stefan O'Rear, 30-Oct-2014.)

Assertion

Ref	Expression
f1imaeq	|- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) = ( F " D ) <-> C = D ) )

Proof of Theorem f1imaeq

Step	Hyp	Ref	Expression
1		f1imass 5742	. . 3 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) <-> C C_ D ) )
2		f1imass 5742	. . . 4 |- ( ( F : A -1-1-> B /\ ( D C_ A /\ C C_ A ) ) -> ( ( F " D ) C_ ( F " C ) <-> D C_ C ) )
3	2	ancom2s 556	. . 3 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " D ) C_ ( F " C ) <-> D C_ C ) )
4	1, 3	anbi12d 465	. 2 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( ( F " C ) C_ ( F " D ) /\ ( F " D ) C_ ( F " C ) ) <-> ( C C_ D /\ D C_ C ) ) )
5		eqss 3157	. 2 |- ( ( F " C ) = ( F " D ) <-> ( ( F " C ) C_ ( F " D ) /\ ( F " D ) C_ ( F " C ) ) )
6		eqss 3157	. 2 |- ( C = D <-> ( C C_ D /\ D C_ C ) )
7	4, 5, 6	3bitr4g 222	1 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) = ( F " D ) <-> C = D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   C_ wss 3116   "cima 4607   -1-1->wf1 5185

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fv 5196

This theorem is referenced by:  hmeoimaf1o  12964