Theorem f1imaeq 5554

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 5553	. . 3 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) <-> C C_ D ) )
2		f1imass 5553	. . . 4 |- ( ( F : A -1-1-> B /\ ( D C_ A /\ C C_ A ) ) -> ( ( F " D ) C_ ( F " C ) <-> D C_ C ) )
3	2	ancom2s 533	. . 3 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " D ) C_ ( F " C ) <-> D C_ C ) )
4	1, 3	anbi12d 457	. 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 3040	. 2 |- ( ( F " C ) = ( F " D ) <-> ( ( F " C ) C_ ( F " D ) /\ ( F " D ) C_ ( F " C ) ) )
6		eqss 3040	. 2 |- ( C = D <-> ( C C_ D /\ D C_ C ) )
7	4, 5, 6	3bitr4g 221	1 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) = ( F " D ) <-> C = D ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   C_ wss 2999   "cima 4441   -1-1->wf1 5012

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-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fv 5023

This theorem is referenced by: (None)