Theorem f1imaeq 5954

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 5953	. . 3 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) <-> C C_ D ) )
2		f1imass 5953	. . . 4 |- ( ( F : A -1-1-> B /\ ( D C_ A /\ C C_ A ) ) -> ( ( F " D ) C_ ( F " C ) <-> D C_ C ) )
3	2	ancom2s 568	. . 3 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " D ) C_ ( F " C ) <-> D C_ C ) )
4	1, 3	anbi12d 473	. 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 3257	. 2 |- ( ( F " C ) = ( F " D ) <-> ( ( F " C ) C_ ( F " D ) /\ ( F " D ) C_ ( F " C ) ) )
6		eqss 3257	. 2 |- ( C = D <-> ( C C_ D /\ D C_ C ) )
7	4, 5, 6	3bitr4g 223	1 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) = ( F " D ) <-> C = D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   C_ wss 3214   "cima 4757   -1-1->wf1 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 4233  ax-pow 4292  ax-pr 4327

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 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fv 5365

This theorem is referenced by:  hmeoimaf1o  15305