Theorem genipv 7310

Description: Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)

Hypotheses

Ref	Expression
genp.1	|- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )
genp.2	|- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )

Assertion

Ref	Expression
genipv	|- ( ( A e. P. /\ B e. P. ) -> ( A F B ) = <. { q e. Q. | E. r e. ( 1st ` A ) E. s e. ( 1st ` B ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` A ) E. s e. ( 2nd ` B ) q = ( r G s ) } >. )

Distinct variable groups:   x, y, z, q, r, s, A   x, B, y, z, q, r, s   x, w, v, G, y, z, q, r, s

Allowed substitution hints:   A( w, v)   B( w, v)   F( x, y, z, w, v, s, r, q)

Proof of Theorem genipv

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5774	. . . 4 |- ( f = A -> ( f F g ) = ( A F g ) )
2		fveq2 5414	. . . . . . 7 |- ( f = A -> ( 1st ` f ) = ( 1st ` A ) )
3	2	rexeqdv 2631	. . . . . 6 |- ( f = A -> ( E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) <-> E. y e. ( 1st ` A ) E. z e. ( 1st ` g ) x = ( y G z ) ) )
4	3	rabbidv 2670	. . . . 5 |- ( f = A -> { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } = { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` g ) x = ( y G z ) } )
5		fveq2 5414	. . . . . . 7 |- ( f = A -> ( 2nd ` f ) = ( 2nd ` A ) )
6	5	rexeqdv 2631	. . . . . 6 |- ( f = A -> ( E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) <-> E. y e. ( 2nd ` A ) E. z e. ( 2nd ` g ) x = ( y G z ) ) )
7	6	rabbidv 2670	. . . . 5 |- ( f = A -> { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } = { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` g ) x = ( y G z ) } )
8	4, 7	opeq12d 3708	. . . 4 |- ( f = A -> <. { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. = <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. )
9	1, 8	eqeq12d 2152	. . 3 |- ( f = A -> ( ( f F g ) = <. { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. <-> ( A F g ) = <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. ) )
10		oveq2 5775	. . . 4 |- ( g = B -> ( A F g ) = ( A F B ) )
11		fveq2 5414	. . . . . . . 8 |- ( g = B -> ( 1st ` g ) = ( 1st ` B ) )
12	11	rexeqdv 2631	. . . . . . 7 |- ( g = B -> ( E. z e. ( 1st ` g ) x = ( y G z ) <-> E. z e. ( 1st ` B ) x = ( y G z ) ) )
13	12	rexbidv 2436	. . . . . 6 |- ( g = B -> ( E. y e. ( 1st ` A ) E. z e. ( 1st ` g ) x = ( y G z ) <-> E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y G z ) ) )
14	13	rabbidv 2670	. . . . 5 |- ( g = B -> { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` g ) x = ( y G z ) } = { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y G z ) } )
15		fveq2 5414	. . . . . . . 8 |- ( g = B -> ( 2nd ` g ) = ( 2nd ` B ) )
16	15	rexeqdv 2631	. . . . . . 7 |- ( g = B -> ( E. z e. ( 2nd ` g ) x = ( y G z ) <-> E. z e. ( 2nd ` B ) x = ( y G z ) ) )
17	16	rexbidv 2436	. . . . . 6 |- ( g = B -> ( E. y e. ( 2nd ` A ) E. z e. ( 2nd ` g ) x = ( y G z ) <-> E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y G z ) ) )
18	17	rabbidv 2670	. . . . 5 |- ( g = B -> { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` g ) x = ( y G z ) } = { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y G z ) } )
19	14, 18	opeq12d 3708	. . . 4 |- ( g = B -> <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. = <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y G z ) } >. )
20	10, 19	eqeq12d 2152	. . 3 |- ( g = B -> ( ( A F g ) = <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. <-> ( A F B ) = <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y G z ) } >. ) )
21		nqex 7164	. . . . . . 7 |- Q. e. _V
22	21	a1i 9	. . . . . 6 |- ( ( f e. P. /\ g e. P. ) -> Q. e. _V )
23		rabssab 3179	. . . . . . 7 |- { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } C_ { x | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) }
24		prop 7276	. . . . . . . . . . . 12 |- ( f e. P. -> <. ( 1st ` f ) , ( 2nd ` f ) >. e. P. )
25		elprnql 7282	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` f ) , ( 2nd ` f ) >. e. P. /\ y e. ( 1st ` f ) ) -> y e. Q. )
26	24, 25	sylan 281	. . . . . . . . . . 11 |- ( ( f e. P. /\ y e. ( 1st ` f ) ) -> y e. Q. )
27		prop 7276	. . . . . . . . . . . 12 |- ( g e. P. -> <. ( 1st ` g ) , ( 2nd ` g ) >. e. P. )
28		elprnql 7282	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` g ) , ( 2nd ` g ) >. e. P. /\ z e. ( 1st ` g ) ) -> z e. Q. )
29	27, 28	sylan 281	. . . . . . . . . . 11 |- ( ( g e. P. /\ z e. ( 1st ` g ) ) -> z e. Q. )
30		genp.2	. . . . . . . . . . . 12 |- ( ( y e. Q. /\ z e. Q. ) -> ( y G z ) e. Q. )
31		eleq1 2200	. . . . . . . . . . . 12 |- ( x = ( y G z ) -> ( x e. Q. <-> ( y G z ) e. Q. ) )
32	30, 31	syl5ibrcom 156	. . . . . . . . . . 11 |- ( ( y e. Q. /\ z e. Q. ) -> ( x = ( y G z ) -> x e. Q. ) )
33	26, 29, 32	syl2an 287	. . . . . . . . . 10 |- ( ( ( f e. P. /\ y e. ( 1st ` f ) ) /\ ( g e. P. /\ z e. ( 1st ` g ) ) ) -> ( x = ( y G z ) -> x e. Q. ) )
34	33	an4s 577	. . . . . . . . 9 |- ( ( ( f e. P. /\ g e. P. ) /\ ( y e. ( 1st ` f ) /\ z e. ( 1st ` g ) ) ) -> ( x = ( y G z ) -> x e. Q. ) )
35	34	rexlimdvva 2555	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) -> x e. Q. ) )
36	35	abssdv 3166	. . . . . . 7 |- ( ( f e. P. /\ g e. P. ) -> { x | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } C_ Q. )
37	23, 36	sstrid 3103	. . . . . 6 |- ( ( f e. P. /\ g e. P. ) -> { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } C_ Q. )
38	22, 37	ssexd 4063	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } e. _V )
39		rabssab 3179	. . . . . . 7 |- { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } C_ { x | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) }
40		elprnqu 7283	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` f ) , ( 2nd ` f ) >. e. P. /\ y e. ( 2nd ` f ) ) -> y e. Q. )
41	24, 40	sylan 281	. . . . . . . . . . 11 |- ( ( f e. P. /\ y e. ( 2nd ` f ) ) -> y e. Q. )
42		elprnqu 7283	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` g ) , ( 2nd ` g ) >. e. P. /\ z e. ( 2nd ` g ) ) -> z e. Q. )
43	27, 42	sylan 281	. . . . . . . . . . 11 |- ( ( g e. P. /\ z e. ( 2nd ` g ) ) -> z e. Q. )
44	41, 43, 32	syl2an 287	. . . . . . . . . 10 |- ( ( ( f e. P. /\ y e. ( 2nd ` f ) ) /\ ( g e. P. /\ z e. ( 2nd ` g ) ) ) -> ( x = ( y G z ) -> x e. Q. ) )
45	44	an4s 577	. . . . . . . . 9 |- ( ( ( f e. P. /\ g e. P. ) /\ ( y e. ( 2nd ` f ) /\ z e. ( 2nd ` g ) ) ) -> ( x = ( y G z ) -> x e. Q. ) )
46	45	rexlimdvva 2555	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) -> x e. Q. ) )
47	46	abssdv 3166	. . . . . . 7 |- ( ( f e. P. /\ g e. P. ) -> { x | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } C_ Q. )
48	39, 47	sstrid 3103	. . . . . 6 |- ( ( f e. P. /\ g e. P. ) -> { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } C_ Q. )
49	22, 48	ssexd 4063	. . . . 5 |- ( ( f e. P. /\ g e. P. ) -> { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } e. _V )
50		opelxp 4564	. . . . 5 |- ( <. { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. e. ( _V X. _V ) <-> ( { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } e. _V /\ { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } e. _V ) )
51	38, 49, 50	sylanbrc 413	. . . 4 |- ( ( f e. P. /\ g e. P. ) -> <. { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. e. ( _V X. _V ) )
52		fveq2 5414	. . . . . . . 8 |- ( w = f -> ( 1st ` w ) = ( 1st ` f ) )
53	52	rexeqdv 2631	. . . . . . 7 |- ( w = f -> ( E. y e. ( 1st ` w ) E. z e. ( 1st ` v ) x = ( y G z ) <-> E. y e. ( 1st ` f ) E. z e. ( 1st ` v ) x = ( y G z ) ) )
54	53	rabbidv 2670	. . . . . 6 |- ( w = f -> { x e. Q. | E. y e. ( 1st ` w ) E. z e. ( 1st ` v ) x = ( y G z ) } = { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` v ) x = ( y G z ) } )
55		fveq2 5414	. . . . . . . 8 |- ( w = f -> ( 2nd ` w ) = ( 2nd ` f ) )
56	55	rexeqdv 2631	. . . . . . 7 |- ( w = f -> ( E. y e. ( 2nd ` w ) E. z e. ( 2nd ` v ) x = ( y G z ) <-> E. y e. ( 2nd ` f ) E. z e. ( 2nd ` v ) x = ( y G z ) ) )
57	56	rabbidv 2670	. . . . . 6 |- ( w = f -> { x e. Q. | E. y e. ( 2nd ` w ) E. z e. ( 2nd ` v ) x = ( y G z ) } = { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` v ) x = ( y G z ) } )
58	54, 57	opeq12d 3708	. . . . 5 |- ( w = f -> <. { x e. Q. | E. y e. ( 1st ` w ) E. z e. ( 1st ` v ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` w ) E. z e. ( 2nd ` v ) x = ( y G z ) } >. = <. { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` v ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` v ) x = ( y G z ) } >. )
59		fveq2 5414	. . . . . . . . 9 |- ( v = g -> ( 1st ` v ) = ( 1st ` g ) )
60	59	rexeqdv 2631	. . . . . . . 8 |- ( v = g -> ( E. z e. ( 1st ` v ) x = ( y G z ) <-> E. z e. ( 1st ` g ) x = ( y G z ) ) )
61	60	rexbidv 2436	. . . . . . 7 |- ( v = g -> ( E. y e. ( 1st ` f ) E. z e. ( 1st ` v ) x = ( y G z ) <-> E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) ) )
62	61	rabbidv 2670	. . . . . 6 |- ( v = g -> { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` v ) x = ( y G z ) } = { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } )
63		fveq2 5414	. . . . . . . . 9 |- ( v = g -> ( 2nd ` v ) = ( 2nd ` g ) )
64	63	rexeqdv 2631	. . . . . . . 8 |- ( v = g -> ( E. z e. ( 2nd ` v ) x = ( y G z ) <-> E. z e. ( 2nd ` g ) x = ( y G z ) ) )
65	64	rexbidv 2436	. . . . . . 7 |- ( v = g -> ( E. y e. ( 2nd ` f ) E. z e. ( 2nd ` v ) x = ( y G z ) <-> E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) ) )
66	65	rabbidv 2670	. . . . . 6 |- ( v = g -> { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` v ) x = ( y G z ) } = { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } )
67	62, 66	opeq12d 3708	. . . . 5 |- ( v = g -> <. { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` v ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` v ) x = ( y G z ) } >. = <. { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. )
68		genp.1	. . . . . 6 |- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 1st ` w ) /\ z e. ( 1st ` v ) /\ x = ( y G z ) ) } , { x e. Q. | E. y e. Q. E. z e. Q. ( y e. ( 2nd ` w ) /\ z e. ( 2nd ` v ) /\ x = ( y G z ) ) } >. )
69	68	genpdf 7309	. . . . 5 |- F = ( w e. P. , v e. P. |-> <. { x e. Q. | E. y e. ( 1st ` w ) E. z e. ( 1st ` v ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` w ) E. z e. ( 2nd ` v ) x = ( y G z ) } >. )
70	58, 67, 69	ovmpog 5898	. . . 4 |- ( ( f e. P. /\ g e. P. /\ <. { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. e. ( _V X. _V ) ) -> ( f F g ) = <. { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. )
71	51, 70	mpd3an3 1316	. . 3 |- ( ( f e. P. /\ g e. P. ) -> ( f F g ) = <. { x e. Q. | E. y e. ( 1st ` f ) E. z e. ( 1st ` g ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` f ) E. z e. ( 2nd ` g ) x = ( y G z ) } >. )
72	9, 20, 71	vtocl2ga 2749	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) = <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y G z ) } >. )
73		eqeq1 2144	. . . . . 6 |- ( x = q -> ( x = ( y G z ) <-> q = ( y G z ) ) )
74	73	2rexbidv 2458	. . . . 5 |- ( x = q -> ( E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y G z ) <-> E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) q = ( y G z ) ) )
75		oveq1 5774	. . . . . . 7 |- ( y = r -> ( y G z ) = ( r G z ) )
76	75	eqeq2d 2149	. . . . . 6 |- ( y = r -> ( q = ( y G z ) <-> q = ( r G z ) ) )
77		oveq2 5775	. . . . . . 7 |- ( z = s -> ( r G z ) = ( r G s ) )
78	77	eqeq2d 2149	. . . . . 6 |- ( z = s -> ( q = ( r G z ) <-> q = ( r G s ) ) )
79	76, 78	cbvrex2v 2661	. . . . 5 |- ( E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) q = ( y G z ) <-> E. r e. ( 1st ` A ) E. s e. ( 1st ` B ) q = ( r G s ) )
80	74, 79	syl6bb 195	. . . 4 |- ( x = q -> ( E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y G z ) <-> E. r e. ( 1st ` A ) E. s e. ( 1st ` B ) q = ( r G s ) ) )
81	80	cbvrabv 2680	. . 3 |- { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y G z ) } = { q e. Q. | E. r e. ( 1st ` A ) E. s e. ( 1st ` B ) q = ( r G s ) }
82	73	2rexbidv 2458	. . . . 5 |- ( x = q -> ( E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y G z ) <-> E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) q = ( y G z ) ) )
83	76, 78	cbvrex2v 2661	. . . . 5 |- ( E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) q = ( y G z ) <-> E. r e. ( 2nd ` A ) E. s e. ( 2nd ` B ) q = ( r G s ) )
84	82, 83	syl6bb 195	. . . 4 |- ( x = q -> ( E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y G z ) <-> E. r e. ( 2nd ` A ) E. s e. ( 2nd ` B ) q = ( r G s ) ) )
85	84	cbvrabv 2680	. . 3 |- { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y G z ) } = { q e. Q. | E. r e. ( 2nd ` A ) E. s e. ( 2nd ` B ) q = ( r G s ) }
86	81, 85	opeq12i 3705	. 2 |- <. { x e. Q. | E. y e. ( 1st ` A ) E. z e. ( 1st ` B ) x = ( y G z ) } , { x e. Q. | E. y e. ( 2nd ` A ) E. z e. ( 2nd ` B ) x = ( y G z ) } >. = <. { q e. Q. | E. r e. ( 1st ` A ) E. s e. ( 1st ` B ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` A ) E. s e. ( 2nd ` B ) q = ( r G s ) } >.
87	72, 86	syl6eq 2186	1 |- ( ( A e. P. /\ B e. P. ) -> ( A F B ) = <. { q e. Q. | E. r e. ( 1st ` A ) E. s e. ( 1st ` B ) q = ( r G s ) } , { q e. Q. | E. r e. ( 2nd ` A ) E. s e. ( 2nd ` B ) q = ( r G s ) } >. )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 962   = wceq 1331   e. wcel 1480   {cab 2123   E.wrex 2415   {crab 2418   _Vcvv 2681   <.cop 3525   X. cxp 4532   ` cfv 5118  (class class class)co 5767   e. cmpo 5769   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   P.cnp 7092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-qs 6428  df-ni 7105  df-nqqs 7149  df-inp 7267

This theorem is referenced by:  genpelvl  7313  genpelvu  7314  plpvlu  7339  mpvlu  7340