Theorem fovcld 6000

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 998	. 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 5999	. . . . 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 1020	. 2 |- ( ( ph /\ A e. R /\ B e. S ) -> A. x e. R A. y e. S ( x F y ) e. C )
7		oveq1 5902	. . . 4 |- ( x = A -> ( x F y ) = ( A F y ) )
8	7	eleq1d 2258	. . 3 |- ( x = A -> ( ( x F y ) e. C <-> ( A F y ) e. C ) )
9		oveq2 5903	. . . 4 |- ( y = B -> ( A F y ) = ( A F B ) )
10	9	eleq1d 2258	. . 3 |- ( y = B -> ( ( A F y ) e. C <-> ( A F B ) e. C ) )
11	8, 10	rspc2v 2869	. 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 980   = wceq 1364   e. wcel 2160   A.wral 2468   X. cxp 4642   Fn wfn 5230   -->wf 5231  (class class class)co 5895

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-ov 5898

This theorem is referenced by:  fovcl  6001  imasrng  13307