Theorem funimass3 5304

Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5303 would be the special case of A being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)

Assertion

Ref	Expression
funimass3	|- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )

Proof of Theorem funimass3

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimass4 5245	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )
2		ssel 2993	. . . . . 6 |- ( A C_ dom F -> ( x e. A -> x e. dom F ) )
3		fvimacnv 5303	. . . . . . 7 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) )
4	3	ex 113	. . . . . 6 |- ( Fun F -> ( x e. dom F -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) ) )
5	2, 4	syl9r 72	. . . . 5 |- ( Fun F -> ( A C_ dom F -> ( x e. A -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) ) ) )
6	5	imp31 252	. . . 4 |- ( ( ( Fun F /\ A C_ dom F ) /\ x e. A ) -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) )
7	6	ralbidva 2364	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( A. x e. A ( F ` x ) e. B <-> A. x e. A x e. ( `' F " B ) ) )
8	1, 7	bitrd 186	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A x e. ( `' F " B ) ) )
9		dfss3 2989	. 2 |- ( A C_ ( `' F " B ) <-> A. x e. A x e. ( `' F " B ) )
10	8, 9	syl6bbr 196	1 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1433   A.wral 2348   C_ wss 2973   `'ccnv 4362   dom cdm 4363   "cima 4366   Fun wfun 4916   ` cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930

This theorem is referenced by:  funimass5  5305  funconstss  5306  fimacnv  5317