Theorem f1imapss 6523

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

Assertion

Ref	Expression
f1imapss	|- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C. ( F " D ) <-> C C. D ) )

Proof of Theorem f1imapss

Step	Hyp	Ref	Expression
1		f1imass 6521	. . 3 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) <-> C C_ D ) )
2		f1imaeq 6522	. . . 4 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) = ( F " D ) <-> C = D ) )
3	2	notbid 308	. . 3 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( -. ( F " C ) = ( F " D ) <-> -. C = D ) )
4	1, 3	anbi12d 747	. 2 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( ( F " C ) C_ ( F " D ) /\ -. ( F " C ) = ( F " D ) ) <-> ( C C_ D /\ -. C = D ) ) )
5		dfpss2 3692	. 2 |- ( ( F " C ) C. ( F " D ) <-> ( ( F " C ) C_ ( F " D ) /\ -. ( F " C ) = ( F " D ) ) )
6		dfpss2 3692	. 2 |- ( C C. D <-> ( C C_ D /\ -. C = D ) )
7	4, 5, 6	3bitr4g 303	1 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C. ( F " D ) <-> C C. D ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574   C. wpss 3575   "cima 5117   -1-1->wf1 5885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fv 5896

This theorem is referenced by:  fin4en1  9131