Theorem f1imass 5898

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 535	. . . . . . 7 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> C C_ A )
2	1	sseld 3223	. . . . . 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 528	. . . . . . . . 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 3223	. . . . . . . 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 533	. . . . . . . . 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 110	. . . . . . . . 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 1086	. . . . . . . . . 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 1227	. . . . . . . . 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 5897	. . . . . . . . 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 1271	. . . . . . . 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 1087	. . . . . . . . . 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 1227	. . . . . . . . 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 5897	. . . . . . . . 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 1271	. . . . . . . 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 202	. . . . . . 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 115	. . . . . 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 45	. . . . 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 50	. . . 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 3230	. . 3 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> C C_ D )
20	19	ex 115	. 2 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) -> C C_ D ) )
21		imass2 5104	. 2 |- ( C C_ D -> ( F " C ) C_ ( F " D ) )
22	20, 21	impbid1 142	1 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) <-> C C_ D ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   C_ wss 3197   "cima 4722   -1-1->wf1 5315   ` cfv 5318

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fv 5326

This theorem is referenced by:  f1imaeq  5899