Theorem f1imaeq 5669

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 5668	. . 3 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) <-> C C_ D ) )
2		f1imass 5668	. . . 4 |- ( ( F : A -1-1-> B /\ ( D C_ A /\ C C_ A ) ) -> ( ( F " D ) C_ ( F " C ) <-> D C_ C ) )
3	2	ancom2s 555	. . 3 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " D ) C_ ( F " C ) <-> D C_ C ) )
4	1, 3	anbi12d 464	. 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 3107	. 2 |- ( ( F " C ) = ( F " D ) <-> ( ( F " C ) C_ ( F " D ) /\ ( F " D ) C_ ( F " C ) ) )
6		eqss 3107	. 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 1331   C_ wss 3066   "cima 4537   -1-1->wf1 5115

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fv 5126

This theorem is referenced by:  hmeoimaf1o  12472