Theorem funimass3 5584

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 5583 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 5520	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )
2		ssel 3122	. . . . . 6 |- ( A C_ dom F -> ( x e. A -> x e. dom F ) )
3		fvimacnv 5583	. . . . . . 7 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) )
4	3	ex 114	. . . . . 6 |- ( Fun F -> ( x e. dom F -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) ) )
5	2, 4	syl9r 73	. . . . 5 |- ( Fun F -> ( A C_ dom F -> ( x e. A -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) ) ) )
6	5	imp31 254	. . . 4 |- ( ( ( Fun F /\ A C_ dom F ) /\ x e. A ) -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) )
7	6	ralbidva 2453	. . 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 187	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A x e. ( `' F " B ) ) )
9		dfss3 3118	. 2 |- ( A C_ ( `' F " B ) <-> A. x e. A x e. ( `' F " B ) )
10	8, 9	bitr4di 197	1 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2128   A.wral 2435   C_ wss 3102   `'ccnv 4586   dom cdm 4587   "cima 4590   Fun wfun 5165   ` cfv 5171

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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-fv 5179

This theorem is referenced by:  funimass5  5585  funconstss  5586  fimacnv  5597  iscnp3  12645  cnpnei  12661  cncnp  12672