Theorem f1imass 6521

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

Assertion

Ref	Expression
f1imass	|- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) <-> C C_ D ) )

Proof of Theorem f1imass

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplrl 800	. . . . . . 7 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> C C_ A )
2	1	sseld 3602	. . . . . 6 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> ( a e. C -> a e. A ) )
3		simplr 792	. . . . . . . . 9 |- ( ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) /\ a e. A ) -> ( F " C ) C_ ( F " D ) )
4	3	sseld 3602	. . . . . . . 8 |- ( ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) /\ a e. A ) -> ( ( F ` a ) e. ( F " C ) -> ( F ` a ) e. ( F " D ) ) )
5		simplll 798	. . . . . . . . 9 |- ( ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) /\ a e. A ) -> F : A -1-1-> B )
6		simpr 477	. . . . . . . . 9 |- ( ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) /\ a e. A ) -> a e. A )
7		simp1rl 1126	. . . . . . . . . 10 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) /\ a e. A ) -> C C_ A )
8	7	3expa 1265	. . . . . . . . 9 |- ( ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) /\ a e. A ) -> C C_ A )
9		f1elima 6520	. . . . . . . . 9 |- ( ( F : A -1-1-> B /\ a e. A /\ C C_ A ) -> ( ( F ` a ) e. ( F " C ) <-> a e. C ) )
10	5, 6, 8, 9	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) /\ a e. A ) -> ( ( F ` a ) e. ( F " C ) <-> a e. C ) )
11		simp1rr 1127	. . . . . . . . . 10 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) /\ a e. A ) -> D C_ A )
12	11	3expa 1265	. . . . . . . . 9 |- ( ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) /\ a e. A ) -> D C_ A )
13		f1elima 6520	. . . . . . . . 9 |- ( ( F : A -1-1-> B /\ a e. A /\ D C_ A ) -> ( ( F ` a ) e. ( F " D ) <-> a e. D ) )
14	5, 6, 12, 13	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) /\ a e. A ) -> ( ( F ` a ) e. ( F " D ) <-> a e. D ) )
15	4, 10, 14	3imtr3d 282	. . . . . . 7 |- ( ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) /\ a e. A ) -> ( a e. C -> a e. D ) )
16	15	ex 450	. . . . . 6 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> ( a e. A -> ( a e. C -> a e. D ) ) )
17	2, 16	syld 47	. . . . 5 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> ( a e. C -> ( a e. C -> a e. D ) ) )
18	17	pm2.43d 53	. . . 4 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> ( a e. C -> a e. D ) )
19	18	ssrdv 3609	. . 3 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> C C_ D )
20	19	ex 450	. 2 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) -> C C_ D ) )
21		imass2 5501	. 2 |- ( C C_ D -> ( F " C ) C_ ( F " D ) )
22	20, 21	impbid1 215	1 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) <-> C C_ D ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   C_ wss 3574   "cima 5117   -1-1->wf1 5885   ` cfv 5888

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-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-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:  f1imaeq  6522  f1imapss  6523  enfin2i  9143  tsmsf1o  21948