Theorem catcxpccl 16847

Description: The category of categories for a weak universe is closed under the product category operation. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
catcxpccl.c	|- C = (CatCat ` U )
catcxpccl.b	|- B = ( Base ` C )
catcxpccl.o	|- T = ( X X.c Y )
catcxpccl.u	|- ( ph -> U e. WUni )
catcxpccl.1	|- ( ph -> om e. U )
catcxpccl.x	|- ( ph -> X e. B )
catcxpccl.y	|- ( ph -> Y e. B )

Assertion

Ref	Expression
catcxpccl	|- ( ph -> T e. B )

Proof of Theorem catcxpccl

Dummy variables f g u v x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		catcxpccl.o	. . . . 5 |- T = ( X X.c Y )
2		eqid 2622	. . . . 5 |- ( Base ` X ) = ( Base ` X )
3		eqid 2622	. . . . 5 |- ( Base ` Y ) = ( Base ` Y )
4		eqid 2622	. . . . 5 |- ( Hom ` X ) = ( Hom ` X )
5		eqid 2622	. . . . 5 |- ( Hom ` Y ) = ( Hom ` Y )
6		eqid 2622	. . . . 5 |- (comp ` X ) = (comp ` X )
7		eqid 2622	. . . . 5 |- (comp ` Y ) = (comp ` Y )
8		catcxpccl.x	. . . . 5 |- ( ph -> X e. B )
9		catcxpccl.y	. . . . 5 |- ( ph -> Y e. B )
10		eqidd 2623	. . . . 5 |- ( ph -> ( ( Base ` X ) X. ( Base ` Y ) ) = ( ( Base ` X ) X. ( Base ` Y ) ) )
11	1, 2, 3	xpcbas 16818	. . . . . . 7 |- ( ( Base ` X ) X. ( Base ` Y ) ) = ( Base ` T )
12		eqid 2622	. . . . . . 7 |- ( Hom ` T ) = ( Hom ` T )
13	1, 11, 4, 5, 12	xpchomfval 16819	. . . . . 6 |- ( Hom ` T ) = ( u e. ( ( Base ` X ) X. ( Base ` Y ) ) , v e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) )
14	13	a1i 11	. . . . 5 |- ( ph -> ( Hom ` T ) = ( u e. ( ( Base ` X ) X. ( Base ` Y ) ) , v e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) ) )
15		eqidd 2623	. . . . 5 |- ( ph -> ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) = ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) )
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15	xpcval 16817	. . . 4 |- ( ph -> T = { <. ( Base ` ndx ) , ( ( Base ` X ) X. ( Base ` Y ) ) >. , <. ( Hom ` ndx ) , ( Hom ` T ) >. , <. (comp ` ndx ) , ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } )
17		catcxpccl.u	. . . . 5 |- ( ph -> U e. WUni )
18		df-base 15863	. . . . . . 7 |- Base = Slot 1
19		catcxpccl.1	. . . . . . . 8 |- ( ph -> om e. U )
20	17, 19	wunndx 15878	. . . . . . 7 |- ( ph -> ndx e. U )
21	18, 17, 20	wunstr 15881	. . . . . 6 |- ( ph -> ( Base ` ndx ) e. U )
22		inss1 3833	. . . . . . . . 9 |- ( U i^i Cat ) C_ U
23		catcxpccl.c	. . . . . . . . . . 11 |- C = (CatCat ` U )
24		catcxpccl.b	. . . . . . . . . . 11 |- B = ( Base ` C )
25	23, 24, 17	catcbas 16747	. . . . . . . . . 10 |- ( ph -> B = ( U i^i Cat ) )
26	8, 25	eleqtrd 2703	. . . . . . . . 9 |- ( ph -> X e. ( U i^i Cat ) )
27	22, 26	sseldi 3601	. . . . . . . 8 |- ( ph -> X e. U )
28	18, 17, 27	wunstr 15881	. . . . . . 7 |- ( ph -> ( Base ` X ) e. U )
29	9, 25	eleqtrd 2703	. . . . . . . . 9 |- ( ph -> Y e. ( U i^i Cat ) )
30	22, 29	sseldi 3601	. . . . . . . 8 |- ( ph -> Y e. U )
31	18, 17, 30	wunstr 15881	. . . . . . 7 |- ( ph -> ( Base ` Y ) e. U )
32	17, 28, 31	wunxp 9546	. . . . . 6 |- ( ph -> ( ( Base ` X ) X. ( Base ` Y ) ) e. U )
33	17, 21, 32	wunop 9544	. . . . 5 |- ( ph -> <. ( Base ` ndx ) , ( ( Base ` X ) X. ( Base ` Y ) ) >. e. U )
34		df-hom 15966	. . . . . . 7 |- Hom = Slot ; 1 4
35	34, 17, 20	wunstr 15881	. . . . . 6 |- ( ph -> ( Hom ` ndx ) e. U )
36	17, 32, 32	wunxp 9546	. . . . . . . 8 |- ( ph -> ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) e. U )
37	34, 17, 27	wunstr 15881	. . . . . . . . . . . 12 |- ( ph -> ( Hom ` X ) e. U )
38	17, 37	wunrn 9551	. . . . . . . . . . 11 |- ( ph -> ran ( Hom ` X ) e. U )
39	17, 38	wununi 9528	. . . . . . . . . 10 |- ( ph -> U. ran ( Hom ` X ) e. U )
40	34, 17, 30	wunstr 15881	. . . . . . . . . . . 12 |- ( ph -> ( Hom ` Y ) e. U )
41	17, 40	wunrn 9551	. . . . . . . . . . 11 |- ( ph -> ran ( Hom ` Y ) e. U )
42	17, 41	wununi 9528	. . . . . . . . . 10 |- ( ph -> U. ran ( Hom ` Y ) e. U )
43	17, 39, 42	wunxp 9546	. . . . . . . . 9 |- ( ph -> ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) ) e. U )
44	17, 43	wunpw 9529	. . . . . . . 8 |- ( ph -> ~P ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) ) e. U )
45		ovssunirn 6681	. . . . . . . . . . . . 13 |- ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) C_ U. ran ( Hom ` X )
46		ovssunirn 6681	. . . . . . . . . . . . 13 |- ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) C_ U. ran ( Hom ` Y )
47		xpss12 5225	. . . . . . . . . . . . 13 |- ( ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) C_ U. ran ( Hom ` X ) /\ ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) C_ U. ran ( Hom ` Y ) ) -> ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) C_ ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) ) )
48	45, 46, 47	mp2an 708	. . . . . . . . . . . 12 |- ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) C_ ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) )
49		ovex 6678	. . . . . . . . . . . . . 14 |- ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) e. _V
50		ovex 6678	. . . . . . . . . . . . . 14 |- ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) e. _V
51	49, 50	xpex 6962	. . . . . . . . . . . . 13 |- ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) e. _V
52	51	elpw 4164	. . . . . . . . . . . 12 |- ( ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) e. ~P ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) ) <-> ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) C_ ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) ) )
53	48, 52	mpbir 221	. . . . . . . . . . 11 |- ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) e. ~P ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) )
54	53	rgen2w 2925	. . . . . . . . . 10 |- A. u e. ( ( Base ` X ) X. ( Base ` Y ) ) A. v e. ( ( Base ` X ) X. ( Base ` Y ) ) ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) e. ~P ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) )
55		eqid 2622	. . . . . . . . . . 11 |- ( u e. ( ( Base ` X ) X. ( Base ` Y ) ) , v e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) ) = ( u e. ( ( Base ` X ) X. ( Base ` Y ) ) , v e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) )
56	55	fmpt2 7237	. . . . . . . . . 10 |- ( A. u e. ( ( Base ` X ) X. ( Base ` Y ) ) A. v e. ( ( Base ` X ) X. ( Base ` Y ) ) ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) e. ~P ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) ) <-> ( u e. ( ( Base ` X ) X. ( Base ` Y ) ) , v e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) ) : ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) --> ~P ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) ) )
57	54, 56	mpbi 220	. . . . . . . . 9 |- ( u e. ( ( Base ` X ) X. ( Base ` Y ) ) , v e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) ) : ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) --> ~P ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) )
58	57	a1i 11	. . . . . . . 8 |- ( ph -> ( u e. ( ( Base ` X ) X. ( Base ` Y ) ) , v e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) ) : ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) --> ~P ( U. ran ( Hom ` X ) X. U. ran ( Hom ` Y ) ) )
59	17, 36, 44, 58	wunf 9549	. . . . . . 7 |- ( ph -> ( u e. ( ( Base ` X ) X. ( Base ` Y ) ) , v e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( ( ( 1st ` u ) ( Hom ` X ) ( 1st ` v ) ) X. ( ( 2nd ` u ) ( Hom ` Y ) ( 2nd ` v ) ) ) ) e. U )
60	13, 59	syl5eqel 2705	. . . . . 6 |- ( ph -> ( Hom ` T ) e. U )
61	17, 35, 60	wunop 9544	. . . . 5 |- ( ph -> <. ( Hom ` ndx ) , ( Hom ` T ) >. e. U )
62		df-cco 15967	. . . . . . 7 |- comp = Slot ; 1 5
63	62, 17, 20	wunstr 15881	. . . . . 6 |- ( ph -> (comp ` ndx ) e. U )
64	17, 36, 32	wunxp 9546	. . . . . . 7 |- ( ph -> ( ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) e. U )
65	62, 17, 27	wunstr 15881	. . . . . . . . . . . . . 14 |- ( ph -> (comp ` X ) e. U )
66	17, 65	wunrn 9551	. . . . . . . . . . . . 13 |- ( ph -> ran (comp ` X ) e. U )
67	17, 66	wununi 9528	. . . . . . . . . . . 12 |- ( ph -> U. ran (comp ` X ) e. U )
68	17, 67	wunrn 9551	. . . . . . . . . . 11 |- ( ph -> ran U. ran (comp ` X ) e. U )
69	17, 68	wununi 9528	. . . . . . . . . 10 |- ( ph -> U. ran U. ran (comp ` X ) e. U )
70	17, 69	wunpw 9529	. . . . . . . . 9 |- ( ph -> ~P U. ran U. ran (comp ` X ) e. U )
71	62, 17, 30	wunstr 15881	. . . . . . . . . . . . . 14 |- ( ph -> (comp ` Y ) e. U )
72	17, 71	wunrn 9551	. . . . . . . . . . . . 13 |- ( ph -> ran (comp ` Y ) e. U )
73	17, 72	wununi 9528	. . . . . . . . . . . 12 |- ( ph -> U. ran (comp ` Y ) e. U )
74	17, 73	wunrn 9551	. . . . . . . . . . 11 |- ( ph -> ran U. ran (comp ` Y ) e. U )
75	17, 74	wununi 9528	. . . . . . . . . 10 |- ( ph -> U. ran U. ran (comp ` Y ) e. U )
76	17, 75	wunpw 9529	. . . . . . . . 9 |- ( ph -> ~P U. ran U. ran (comp ` Y ) e. U )
77	17, 70, 76	wunxp 9546	. . . . . . . 8 |- ( ph -> ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) e. U )
78	17, 60	wunrn 9551	. . . . . . . . . 10 |- ( ph -> ran ( Hom ` T ) e. U )
79	17, 78	wununi 9528	. . . . . . . . 9 |- ( ph -> U. ran ( Hom ` T ) e. U )
80	17, 79, 79	wunxp 9546	. . . . . . . 8 |- ( ph -> ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) e. U )
81	17, 77, 80	wunpm 9547	. . . . . . 7 |- ( ph -> ( ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) ^pm ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) ) e. U )
82		fvex 6201	. . . . . . . . . . . . . . . . 17 |- (comp ` X ) e. _V
83	82	rnex 7100	. . . . . . . . . . . . . . . 16 |- ran (comp ` X ) e. _V
84	83	uniex 6953	. . . . . . . . . . . . . . 15 |- U. ran (comp ` X ) e. _V
85	84	rnex 7100	. . . . . . . . . . . . . 14 |- ran U. ran (comp ` X ) e. _V
86	85	uniex 6953	. . . . . . . . . . . . 13 |- U. ran U. ran (comp ` X ) e. _V
87	86	pwex 4848	. . . . . . . . . . . 12 |- ~P U. ran U. ran (comp ` X ) e. _V
88		fvex 6201	. . . . . . . . . . . . . . . . 17 |- (comp ` Y ) e. _V
89	88	rnex 7100	. . . . . . . . . . . . . . . 16 |- ran (comp ` Y ) e. _V
90	89	uniex 6953	. . . . . . . . . . . . . . 15 |- U. ran (comp ` Y ) e. _V
91	90	rnex 7100	. . . . . . . . . . . . . 14 |- ran U. ran (comp ` Y ) e. _V
92	91	uniex 6953	. . . . . . . . . . . . 13 |- U. ran U. ran (comp ` Y ) e. _V
93	92	pwex 4848	. . . . . . . . . . . 12 |- ~P U. ran U. ran (comp ` Y ) e. _V
94	87, 93	xpex 6962	. . . . . . . . . . 11 |- ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) e. _V
95		fvex 6201	. . . . . . . . . . . . . 14 |- ( Hom ` T ) e. _V
96	95	rnex 7100	. . . . . . . . . . . . 13 |- ran ( Hom ` T ) e. _V
97	96	uniex 6953	. . . . . . . . . . . 12 |- U. ran ( Hom ` T ) e. _V
98	97, 97	xpex 6962	. . . . . . . . . . 11 |- ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) e. _V
99		ovssunirn 6681	. . . . . . . . . . . . . . . 16 |- ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) C_ U. ran ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) )
100		ovssunirn 6681	. . . . . . . . . . . . . . . . 17 |- ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) C_ U. ran (comp ` X )
101		rnss 5354	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) C_ U. ran (comp ` X ) -> ran ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) C_ ran U. ran (comp ` X ) )
102		uniss 4458	. . . . . . . . . . . . . . . . 17 |- ( ran ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) C_ ran U. ran (comp ` X ) -> U. ran ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) C_ U. ran U. ran (comp ` X ) )
103	100, 101, 102	mp2b 10	. . . . . . . . . . . . . . . 16 |- U. ran ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) C_ U. ran U. ran (comp ` X )
104	99, 103	sstri 3612	. . . . . . . . . . . . . . 15 |- ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) C_ U. ran U. ran (comp ` X )
105		ovex 6678	. . . . . . . . . . . . . . . 16 |- ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) e. _V
106	105	elpw 4164	. . . . . . . . . . . . . . 15 |- ( ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) e. ~P U. ran U. ran (comp ` X ) <-> ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) C_ U. ran U. ran (comp ` X ) )
107	104, 106	mpbir 221	. . . . . . . . . . . . . 14 |- ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) e. ~P U. ran U. ran (comp ` X )
108		ovssunirn 6681	. . . . . . . . . . . . . . . 16 |- ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) C_ U. ran ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) )
109		ovssunirn 6681	. . . . . . . . . . . . . . . . 17 |- ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) C_ U. ran (comp ` Y )
110		rnss 5354	. . . . . . . . . . . . . . . . 17 |- ( ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) C_ U. ran (comp ` Y ) -> ran ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) C_ ran U. ran (comp ` Y ) )
111		uniss 4458	. . . . . . . . . . . . . . . . 17 |- ( ran ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) C_ ran U. ran (comp ` Y ) -> U. ran ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) C_ U. ran U. ran (comp ` Y ) )
112	109, 110, 111	mp2b 10	. . . . . . . . . . . . . . . 16 |- U. ran ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) C_ U. ran U. ran (comp ` Y )
113	108, 112	sstri 3612	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) C_ U. ran U. ran (comp ` Y )
114		ovex 6678	. . . . . . . . . . . . . . . 16 |- ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) e. _V
115	114	elpw 4164	. . . . . . . . . . . . . . 15 |- ( ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) e. ~P U. ran U. ran (comp ` Y ) <-> ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) C_ U. ran U. ran (comp ` Y ) )
116	113, 115	mpbir 221	. . . . . . . . . . . . . 14 |- ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) e. ~P U. ran U. ran (comp ` Y )
117		opelxpi 5148	. . . . . . . . . . . . . 14 |- ( ( ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) e. ~P U. ran U. ran (comp ` X ) /\ ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) e. ~P U. ran U. ran (comp ` Y ) ) -> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. e. ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) )
118	107, 116, 117	mp2an 708	. . . . . . . . . . . . 13 |- <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. e. ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) )
119	118	rgen2w 2925	. . . . . . . . . . . 12 |- A. g e. ( ( 2nd ` x ) ( Hom ` T ) y ) A. f e. ( ( Hom ` T ) ` x ) <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. e. ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) )
120		eqid 2622	. . . . . . . . . . . . 13 |- ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) = ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. )
121	120	fmpt2 7237	. . . . . . . . . . . 12 |- ( A. g e. ( ( 2nd ` x ) ( Hom ` T ) y ) A. f e. ( ( Hom ` T ) ` x ) <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. e. ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) <-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) : ( ( ( 2nd ` x ) ( Hom ` T ) y ) X. ( ( Hom ` T ) ` x ) ) --> ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) )
122	119, 121	mpbi 220	. . . . . . . . . . 11 |- ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) : ( ( ( 2nd ` x ) ( Hom ` T ) y ) X. ( ( Hom ` T ) ` x ) ) --> ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) )
123		ovssunirn 6681	. . . . . . . . . . . 12 |- ( ( 2nd ` x ) ( Hom ` T ) y ) C_ U. ran ( Hom ` T )
124		fvssunirn 6217	. . . . . . . . . . . 12 |- ( ( Hom ` T ) ` x ) C_ U. ran ( Hom ` T )
125		xpss12 5225	. . . . . . . . . . . 12 |- ( ( ( ( 2nd ` x ) ( Hom ` T ) y ) C_ U. ran ( Hom ` T ) /\ ( ( Hom ` T ) ` x ) C_ U. ran ( Hom ` T ) ) -> ( ( ( 2nd ` x ) ( Hom ` T ) y ) X. ( ( Hom ` T ) ` x ) ) C_ ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) )
126	123, 124, 125	mp2an 708	. . . . . . . . . . 11 |- ( ( ( 2nd ` x ) ( Hom ` T ) y ) X. ( ( Hom ` T ) ` x ) ) C_ ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) )
127		elpm2r 7875	. . . . . . . . . . 11 |- ( ( ( ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) e. _V /\ ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) e. _V ) /\ ( ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) : ( ( ( 2nd ` x ) ( Hom ` T ) y ) X. ( ( Hom ` T ) ` x ) ) --> ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) /\ ( ( ( 2nd ` x ) ( Hom ` T ) y ) X. ( ( Hom ` T ) ` x ) ) C_ ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) ) ) -> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) e. ( ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) ^pm ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) ) )
128	94, 98, 122, 126, 127	mp4an 709	. . . . . . . . . 10 |- ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) e. ( ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) ^pm ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) )
129	128	rgen2w 2925	. . . . . . . . 9 |- A. x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) A. y e. ( ( Base ` X ) X. ( Base ` Y ) ) ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) e. ( ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) ^pm ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) )
130		eqid 2622	. . . . . . . . . 10 |- ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) = ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) )
131	130	fmpt2 7237	. . . . . . . . 9 |- ( A. x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) A. y e. ( ( Base ` X ) X. ( Base ` Y ) ) ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) e. ( ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) ^pm ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) ) <-> ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) : ( ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) --> ( ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) ^pm ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) ) )
132	129, 131	mpbi 220	. . . . . . . 8 |- ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) : ( ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) --> ( ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) ^pm ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) )
133	132	a1i 11	. . . . . . 7 |- ( ph -> ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) : ( ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) --> ( ( ~P U. ran U. ran (comp ` X ) X. ~P U. ran U. ran (comp ` Y ) ) ^pm ( U. ran ( Hom ` T ) X. U. ran ( Hom ` T ) ) ) )
134	17, 64, 81, 133	wunf 9549	. . . . . 6 |- ( ph -> ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) e. U )
135	17, 63, 134	wunop 9544	. . . . 5 |- ( ph -> <. (comp ` ndx ) , ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. e. U )
136	17, 33, 61, 135	wuntp 9533	. . . 4 |- ( ph -> { <. ( Base ` ndx ) , ( ( Base ` X ) X. ( Base ` Y ) ) >. , <. ( Hom ` ndx ) , ( Hom ` T ) >. , <. (comp ` ndx ) , ( x e. ( ( ( Base ` X ) X. ( Base ` Y ) ) X. ( ( Base ` X ) X. ( Base ` Y ) ) ) , y e. ( ( Base ` X ) X. ( Base ` Y ) ) |-> ( g e. ( ( 2nd ` x ) ( Hom ` T ) y ) , f e. ( ( Hom ` T ) ` x ) |-> <. ( ( 1st ` g ) ( <. ( 1st ` ( 1st ` x ) ) , ( 1st ` ( 2nd ` x ) ) >. (comp ` X ) ( 1st ` y ) ) ( 1st ` f ) ) , ( ( 2nd ` g ) ( <. ( 2nd ` ( 1st ` x ) ) , ( 2nd ` ( 2nd ` x ) ) >. (comp ` Y ) ( 2nd ` y ) ) ( 2nd ` f ) ) >. ) ) >. } e. U )
137	16, 136	eqeltrd 2701	. . 3 |- ( ph -> T e. U )
138		inss2 3834	. . . . 5 |- ( U i^i Cat ) C_ Cat
139	138, 26	sseldi 3601	. . . 4 |- ( ph -> X e. Cat )
140	138, 29	sseldi 3601	. . . 4 |- ( ph -> Y e. Cat )
141	1, 139, 140	xpccat 16830	. . 3 |- ( ph -> T e. Cat )
142	137, 141	elind 3798	. 2 |- ( ph -> T e. ( U i^i Cat ) )
143	142, 25	eleqtrrd 2704	1 |- ( ph -> T e. B )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   {ctp 4181   <.cop 4183   U.cuni 4436   X. cxp 5112   ran crn 5115   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   omcom 7065   1stc1st 7166   2ndc2nd 7167   ^pm cpm 7858  WUnicwun 9522   1c1 9937   4c4 11072   5c5 11073  ;cdc 11493   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325  CatCatccatc 16744   X._c cxpc 16808

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-wun 9524  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-rq 9739  df-ltnq 9740  df-np 9803  df-plp 9805  df-ltp 9807  df-enr 9877  df-nr 9878  df-c 9942  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-catc 16745  df-xpc 16812

This theorem is referenced by: (None)