Theorem funimass3 5793

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 5792 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 5726	. . 3 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A ( F ` x ) e. B ) )
2		ssel 3231	. . . . . 6 |- ( A C_ dom F -> ( x e. A -> x e. dom F ) )
3		fvimacnv 5792	. . . . . . 7 |- ( ( Fun F /\ x e. dom F ) -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) )
4	3	ex 115	. . . . . 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 256	. . . 4 |- ( ( ( Fun F /\ A C_ dom F ) /\ x e. A ) -> ( ( F ` x ) e. B <-> x e. ( `' F " B ) ) )
7	6	ralbidva 2538	. . 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 188	. 2 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A. x e. A x e. ( `' F " B ) ) )
9		dfss3 3226	. 2 |- ( A C_ ( `' F " B ) <-> A. x e. A x e. ( `' F " B ) )
10	8, 9	bitr4di 198	1 |- ( ( Fun F /\ A C_ dom F ) -> ( ( F " A ) C_ B <-> A C_ ( `' F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2203   A.wral 2520   C_ wss 3210   `'ccnv 4747   dom cdm 4748   "cima 4751   Fun wfun 5345   ` cfv 5351

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2814  df-sbc 3042  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-fv 5359

This theorem is referenced by:  funimass5  5794  funconstss  5795  fimacnv  5805  iscnp3  15060  cnpnei  15076  cncnp  15087