Theorem fovcld 6109

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 1020	. 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 6108	. . . . 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 1042	. 2 |- ( ( ph /\ A e. R /\ B e. S ) -> A. x e. R A. y e. S ( x F y ) e. C )
7		oveq1 6008	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
8	7	eleq1d 2298	. . 3 |- ( x = A -> ( ( x F y ) e. C <-> ( A F y ) e. C ) )
9		oveq2 6009	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
10	9	eleq1d 2298	. . 3 |- ( y = B -> ( ( A F y ) e. C <-> ( A F B ) e. C ) )
11	8, 10	rspc2v 2920	. 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 1002   = wceq 1395   e. wcel 2200   A.wral 2508   X. cxp 4717   Fn wfn 5313   -->wf 5314  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-ov 6004

This theorem is referenced by:  fovcl  6110  imasrng  13919