Theorem f1imass 5675

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 524	. . . . . . 7 |- ( ( ( F : A -1-1-> B /\ ( C C_ A /\ D C_ A ) ) /\ ( F " C ) C_ ( F " D ) ) -> C C_ A )
2	1	sseld 3096	. . . . . 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 519	. . . . . . . . 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 3096	. . . . . . . 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 522	. . . . . . . . 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 1046	. . . . . . . . . 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 1181	. . . . . . . . 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 5674	. . . . . . . . 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 1216	. . . . . . . 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 1047	. . . . . . . . . 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 1181	. . . . . . . . 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 5674	. . . . . . . . 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 1216	. . . . . . . 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 3103	. . 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 4915	. 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 1480   C_ wss 3071   "cima 4542   -1-1->wf1 5120   ` cfv 5123

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fv 5131

This theorem is referenced by:  f1imaeq  5676