Theorem faovcl 41280

Description: Closure law for an operation, analogous to fovcl 6765. (Contributed by Alexander van der Vekens, 26-May-2017.)

Hypothesis

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

Assertion

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

Proof of Theorem faovcl

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

Step	Hyp	Ref	Expression
1		faovcl.1	. . 3 |- F : ( R X. S ) --> C
2		ffnaov 41279	. . . 4 |- ( F : ( R X. S ) --> C <-> ( F Fn ( R X. S ) /\ A. x e. R A. y e. S (( x F y)) e. C ) )
3	2	simprbi 480	. . 3 |- ( F : ( R X. S ) --> C -> A. x e. R A. y e. S (( x F y)) e. C )
4	1, 3	ax-mp 5	. 2 |- A. x e. R A. y e. S (( x F y)) e. C
5		eqidd 2623	. . . . 5 |- ( x = A -> F = F )
6		id 22	. . . . 5 |- ( x = A -> x = A )
7		eqidd 2623	. . . . 5 |- ( x = A -> y = y )
8	5, 6, 7	aoveq123d 41258	. . . 4 |- ( x = A -> (( x F y)) = (( A F y)) )
9	8	eleq1d 2686	. . 3 |- ( x = A -> ( (( x F y)) e. C <-> (( A F y)) e. C ) )
10		eqidd 2623	. . . . 5 |- ( y = B -> F = F )
11		eqidd 2623	. . . . 5 |- ( y = B -> A = A )
12		id 22	. . . . 5 |- ( y = B -> y = B )
13	10, 11, 12	aoveq123d 41258	. . . 4 |- ( y = B -> (( A F y)) = (( A F B)) )
14	13	eleq1d 2686	. . 3 |- ( y = B -> ( (( A F y)) e. C <-> (( A F B)) e. C ) )
15	9, 14	rspc2v 3322	. 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 ) )
16	4, 15	mpi 20	1 |- ( ( A e. R /\ B e. S ) -> (( A F B)) e. C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   X. cxp 5112   Fn wfn 5883   -->wf 5884   ((caov 41195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)