Theorem f1imass 5753

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 530	. . . . . . 7 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> C C_ A )
2	1	sseld 3146	. . . . . 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 525	. . . . . . . . 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 3146	. . . . . . . 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 528	. . . . . . . . 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 109	. . . . . . . . 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 1057	. . . . . . . . . 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 1198	. . . . . . . . 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 5752	. . . . . . . . 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 1233	. . . . . . . 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 1058	. . . . . . . . . 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 1198	. . . . . . . . 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 5752	. . . . . . . . 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 1233	. . . . . . . 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 201	. . . . . . 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 114	. . . . . 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 3153	. . 3 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> C C_ D )
20	19	ex 114	. 2 |- ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) -> ( ( F " C ) C_ ( F " D ) -> C C_ D ) )
21		imass2 4987	. 2 |- ( C C_ D -> ( F " C ) C_ ( F " D ) )
22	20, 21	impbid1 141	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 103   <-> wb 104   e. wcel 2141   C_ wss 3121   "cima 4614   -1-1->wf1 5195   ` cfv 5198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fv 5206

This theorem is referenced by:  f1imaeq  5754