Theorem fovcld 6166

Description: Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)

Hypothesis

Ref	Expression
fovcld.1	|- ( ph -> F : ( R X. S ) --> C )

Assertion

Ref	Expression
fovcld	|- ( ( ph /\ A e. R /\ B e. S ) -> ( A F B ) e. C )

Proof of Theorem fovcld

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		3simpc 1023	. 2 |- ( ( ph /\ A e. R /\ B e. S ) -> ( A e. R /\ B e. S ) )
2		fovcld.1	. . . 4 |- ( ph -> F : ( R X. S ) --> C )
3		ffnov 6165	. . . . 5 |- ( F : ( R X. S ) --> C <-> ( F Fn ( R X. S ) /\ A. x e. R A. y e. S ( x F y ) e. C ) )
4	3	simprbi 275	. . . 4 |- ( F : ( R X. S ) --> C -> A. x e. R A. y e. S ( x F y ) e. C )
5	2, 4	syl 14	. . 3 |- ( ph -> A. x e. R A. y e. S ( x F y ) e. C )
6	5	3ad2ant1 1045	. 2 |- ( ( ph /\ A e. R /\ B e. S ) -> A. x e. R A. y e. S ( x F y ) e. C )
7		oveq1 6065	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
8	7	eleq1d 2303	. . 3 |- ( x = A -> ( ( x F y ) e. C <-> ( A F y ) e. C ) )
9		oveq2 6066	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
10	9	eleq1d 2303	. . 3 |- ( y = B -> ( ( A F y ) e. C <-> ( A F B ) e. C ) )
11	8, 10	rspc2v 2937	. 2 |- ( ( A e. R /\ B e. S ) -> ( A. x e. R A. y e. S ( x F y ) e. C -> ( A F B ) e. C ) )
12	1, 6, 11	sylc 62	1 |- ( ( ph /\ A e. R /\ B e. S ) -> ( A F B ) e. C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   X. cxp 4752   Fn wfn 5352   -->wf 5353  (class class class)co 6058

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 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061

This theorem is referenced by:  fovcl  6167  imasrng  14195