Theorem genipv 7876

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 6092	. . . 4 |- ( f = A -> ( f F g ) = ( A F g ) )
2		fveq2 5695	. . . . . . 7 |- ( f = A -> ( 1st ` f ) = ( 1st ` A ) )
3	2	rexeqdv 2756	. . . . . 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 2810	. . . . 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 5695	. . . . . . 7 |- ( f = A -> ( 2nd ` f ) = ( 2nd ` A ) )
6	5	rexeqdv 2756	. . . . . 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 2810	. . . . 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 3912	. . . 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 2253	. . 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 6093	. . . 4 |- ( g = B -> ( A F g ) = ( A F B ) )
11		fveq2 5695	. . . . . . . 8 |- ( g = B -> ( 1st ` g ) = ( 1st ` B ) )
12	11	rexeqdv 2756	. . . . . . 7 |- ( g = B -> ( E. z e. ( 1st ` g ) x = ( y G z ) <-> E. z e. ( 1st ` B ) x = ( y G z ) ) )
13	12	rexbidv 2551	. . . . . 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 2810	. . . . 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 5695	. . . . . . . 8 |- ( g = B -> ( 2nd ` g ) = ( 2nd ` B ) )
16	15	rexeqdv 2756	. . . . . . 7 |- ( g = B -> ( E. z e. ( 2nd ` g ) x = ( y G z ) <-> E. z e. ( 2nd ` B ) x = ( y G z ) ) )
17	16	rexbidv 2551	. . . . . 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 2810	. . . . 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 3912	. . . 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 2253	. . 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 7730	. . . . . . 7 |- Q. e. _V
22	21	a1i 9	. . . . . 6 |- ( ( f e. P. /\ g e. P. ) -> Q. e. _V )
23		rabssab 3337	. . . . . . 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 7842	. . . . . . . . . . . 12 |- ( f e. P. -> <. ( 1st ` f ) , ( 2nd ` f ) >. e. P. )
25		elprnql 7848	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` f ) , ( 2nd ` f ) >. e. P. /\ y e. ( 1st ` f ) ) -> y e. Q. )
26	24, 25	sylan 283	. . . . . . . . . . 11 |- ( ( f e. P. /\ y e. ( 1st ` f ) ) -> y e. Q. )
27		prop 7842	. . . . . . . . . . . 12 |- ( g e. P. -> <. ( 1st ` g ) , ( 2nd ` g ) >. e. P. )
28		elprnql 7848	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` g ) , ( 2nd ` g ) >. e. P. /\ z e. ( 1st ` g ) ) -> z e. Q. )
29	27, 28	sylan 283	. . . . . . . . . . 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 2301	. . . . . . . . . . . 12 |- ( x = ( y G z ) -> ( x e. Q. <-> ( y G z ) e. Q. ) )
32	30, 31	syl5ibrcom 157	. . . . . . . . . . 11 |- ( ( y e. Q. /\ z e. Q. ) -> ( x = ( y G z ) -> x e. Q. ) )
33	26, 29, 32	syl2an 289	. . . . . . . . . 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 596	. . . . . . . . 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 2676	. . . . . . . 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 3322	. . . . . . 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 3259	. . . . . 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 4273	. . . . 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 3337	. . . . . . 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 7849	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` f ) , ( 2nd ` f ) >. e. P. /\ y e. ( 2nd ` f ) ) -> y e. Q. )
41	24, 40	sylan 283	. . . . . . . . . . 11 |- ( ( f e. P. /\ y e. ( 2nd ` f ) ) -> y e. Q. )
42		elprnqu 7849	. . . . . . . . . . . 12 |- ( ( <. ( 1st ` g ) , ( 2nd ` g ) >. e. P. /\ z e. ( 2nd ` g ) ) -> z e. Q. )
43	27, 42	sylan 283	. . . . . . . . . . 11 |- ( ( g e. P. /\ z e. ( 2nd ` g ) ) -> z e. Q. )
44	41, 43, 32	syl2an 289	. . . . . . . . . 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 596	. . . . . . . . 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 2676	. . . . . . . 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 3322	. . . . . . 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 3259	. . . . . 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 4273	. . . . 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 4804	. . . . 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 421	. . . 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 5695	. . . . . . . 8 |- ( w = f -> ( 1st ` w ) = ( 1st ` f ) )
53	52	rexeqdv 2756	. . . . . . 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 2810	. . . . . 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 5695	. . . . . . . 8 |- ( w = f -> ( 2nd ` w ) = ( 2nd ` f ) )
56	55	rexeqdv 2756	. . . . . . 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 2810	. . . . . 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 3912	. . . . 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 5695	. . . . . . . . 9 |- ( v = g -> ( 1st ` v ) = ( 1st ` g ) )
60	59	rexeqdv 2756	. . . . . . . 8 |- ( v = g -> ( E. z e. ( 1st ` v ) x = ( y G z ) <-> E. z e. ( 1st ` g ) x = ( y G z ) ) )
61	60	rexbidv 2551	. . . . . . 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 2810	. . . . . 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 5695	. . . . . . . . 9 |- ( v = g -> ( 2nd ` v ) = ( 2nd ` g ) )
64	63	rexeqdv 2756	. . . . . . . 8 |- ( v = g -> ( E. z e. ( 2nd ` v ) x = ( y G z ) <-> E. z e. ( 2nd ` g ) x = ( y G z ) ) )
65	64	rexbidv 2551	. . . . . . 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 2810	. . . . . 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 3912	. . . . 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 7875	. . . . 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 6223	. . . 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 1379	. . 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 2891	. 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 2245	. . . . . 6 |- ( x = q -> ( x = ( y G z ) <-> q = ( y G z ) ) )
74	73	2rexbidv 2575	. . . . 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 6092	. . . . . . 7 |- ( y = r -> ( y G z ) = ( r G z ) )
76	75	eqeq2d 2250	. . . . . 6 |- ( y = r -> ( q = ( y G z ) <-> q = ( r G z ) ) )
77		oveq2 6093	. . . . . . 7 |- ( z = s -> ( r G z ) = ( r G s ) )
78	77	eqeq2d 2250	. . . . . 6 |- ( z = s -> ( q = ( r G z ) <-> q = ( r G s ) ) )
79	76, 78	cbvrex2v 2800	. . . . 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	bitrdi 196	. . . 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 2820	. . 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 2575	. . . . 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 2800	. . . . 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	bitrdi 196	. . . 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 2820	. . 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 3909	. 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	eqtrdi 2287	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 ) } >. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   {crab 2532   _Vcvv 2821   <.cop 3712   X. cxp 4772   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373   Q.cnq 7647   P.cnp 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-qs 6813  df-ni 7671  df-nqqs 7715  df-inp 7833

This theorem is used by:  genpelvl  7879  genpelvu  7880  plpvlu  7905  mpvlu  7906