Theorem f1imass 5914

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 537	. . . . . . 7 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> C C_ A )
2	1	sseld 3226	. . . . . 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 529	. . . . . . . . 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 3226	. . . . . . . 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 535	. . . . . . . . 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 1088	. . . . . . . . . 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 1229	. . . . . . . . 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 5913	. . . . . . . . 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 1273	. . . . . . . 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 1089	. . . . . . . . . 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 1229	. . . . . . . . 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 5913	. . . . . . . . 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 1273	. . . . . . . 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 3233	. . 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 5112	. 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 2202   C_ wss 3200   "cima 4728   -1-1->wf1 5323   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fv 5334

This theorem is referenced by:  f1imaeq  5915