Theorem mucex 6134

Description: Cardinal multiplication is a set. (Contributed by SF, 24-Feb-2015.)

Assertion

Ref	Expression
mucex	|- ·c e. _V

Proof of Theorem mucex

Dummy variables a b c d m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-muc 6103	. . 3 |- ·c = (m e. NC , n e. NC |-> {a | E.b e. m E.c e. n a ~~ (b X. c)})
2		elin 3220	. . . . . . . . 9 |- (<.{c}, <.{d}, <.m, n>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c)) <-> (<.{c}, <.{d}, <.m, n>.>.>. e. Ins2 Ins2 S /\ <.{c}, <.{d}, <.m, n>.>.>. e. Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c)))
3		snex 4112	. . . . . . . . . . . 12 |- {d} e. _V
4	3	otelins2 5792	. . . . . . . . . . 11 |- (<.{c}, <.{d}, <.m, n>.>.>. e. Ins2 Ins2 S <-> <.{c}, <.m, n>.>. e. Ins2 S )
5		vex 2863	. . . . . . . . . . . 12 |- m e. _V
6	5	otelins2 5792	. . . . . . . . . . 11 |- (<.{c}, <.m, n>.>. e. Ins2 S <-> <.{c}, n>. e. S )
7		vex 2863	. . . . . . . . . . . 12 |- c e. _V
8		vex 2863	. . . . . . . . . . . 12 |- n e. _V
9	7, 8	opelssetsn 4761	. . . . . . . . . . 11 |- (<.{c}, n>. e. S <-> c e. n)
10	4, 6, 9	3bitri 262	. . . . . . . . . 10 |- (<.{c}, <.{d}, <.m, n>.>.>. e. Ins2 Ins2 S <-> c e. n)
11	8	oqelins4 5795	. . . . . . . . . . 11 |- (<.{c}, <.{d}, <.m, n>.>.>. e. Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c) <-> <.{c}, <.{d}, m>.>. e. (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c))
12		elin 3220	. . . . . . . . . . . . . 14 |- (<.{b}, <.{c}, <.{d}, m>.>.>. e. ( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ )) <-> (<.{b}, <.{c}, <.{d}, m>.>.>. e. Ins2 Ins2 S /\ <.{b}, <.{c}, <.{d}, m>.>.>. e. Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ )))
13		snex 4112	. . . . . . . . . . . . . . . . 17 |- {c} e. _V
14	13	otelins2 5792	. . . . . . . . . . . . . . . 16 |- (<.{b}, <.{c}, <.{d}, m>.>.>. e. Ins2 Ins2 S <-> <.{b}, <.{d}, m>.>. e. Ins2 S )
15	3	otelins2 5792	. . . . . . . . . . . . . . . 16 |- (<.{b}, <.{d}, m>.>. e. Ins2 S <-> <.{b}, m>. e. S )
16		vex 2863	. . . . . . . . . . . . . . . . 17 |- b e. _V
17	16, 5	opelssetsn 4761	. . . . . . . . . . . . . . . 16 |- (<.{b}, m>. e. S <-> b e. m)
18	14, 15, 17	3bitri 262	. . . . . . . . . . . . . . 15 |- (<.{b}, <.{c}, <.{d}, m>.>.>. e. Ins2 Ins2 S <-> b e. m)
19	5	oqelins4 5795	. . . . . . . . . . . . . . . 16 |- (<.{b}, <.{c}, <.{d}, m>.>.>. e. Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) <-> <.{b}, <.{c}, {d}>.>. e. SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))
20		vex 2863	. . . . . . . . . . . . . . . . 17 |- d e. _V
21	16, 7, 20	otsnelsi3 5806	. . . . . . . . . . . . . . . 16 |- (<.{b}, <.{c}, {d}>.>. e. SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) <-> <.b, <.c, d>.>. e. ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))
22		elrn2 4898	. . . . . . . . . . . . . . . . 17 |- (<.b, <.c, d>.>. e. ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) <-> E.a<.a, <.b, <.c, d>.>.>. e. ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))
23		elin 3220	. . . . . . . . . . . . . . . . . . 19 |- (<.a, <.b, <.c, d>.>.>. e. ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) <-> (<.a, <.b, <.c, d>.>.>. e. Ins4 `' Cross /\ <.a, <.b, <.c, d>.>.>. e. Ins2 Ins2 `' ~~ ))
24	20	oqelins4 5795	. . . . . . . . . . . . . . . . . . . . 21 |- (<.a, <.b, <.c, d>.>.>. e. Ins4 `' Cross <-> <.a, <.b, c>.>. e. `' Cross )
25		df-br 4641	. . . . . . . . . . . . . . . . . . . . 21 |- (a`' Cross <.b, c>. <-> <.a, <.b, c>.>. e. `' Cross )
26		brcnv 4893	. . . . . . . . . . . . . . . . . . . . . 22 |- (a`' Cross <.b, c>. <-> <.b, c>. Cross a)
27	16, 7	brcross 5850	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.b, c>. Cross a <-> a = (b X. c))
28	26, 27	bitri 240	. . . . . . . . . . . . . . . . . . . . 21 |- (a`' Cross <.b, c>. <-> a = (b X. c))
29	24, 25, 28	3bitr2i 264	. . . . . . . . . . . . . . . . . . . 20 |- (<.a, <.b, <.c, d>.>.>. e. Ins4 `' Cross <-> a = (b X. c))
30	16	otelins2 5792	. . . . . . . . . . . . . . . . . . . . 21 |- (<.a, <.b, <.c, d>.>.>. e. Ins2 Ins2 `' ~~ <-> <.a, <.c, d>.>. e. Ins2 `' ~~ )
31	7	otelins2 5792	. . . . . . . . . . . . . . . . . . . . . 22 |- (<.a, <.c, d>.>. e. Ins2 `' ~~ <-> <.a, d>. e. `' ~~ )
32		df-br 4641	. . . . . . . . . . . . . . . . . . . . . 22 |- (a`' ~~ d <-> <.a, d>. e. `' ~~ )
33		brcnv 4893	. . . . . . . . . . . . . . . . . . . . . 22 |- (a`' ~~ d <-> d ~~ a)
34	31, 32, 33	3bitr2i 264	. . . . . . . . . . . . . . . . . . . . 21 |- (<.a, <.c, d>.>. e. Ins2 `' ~~ <-> d ~~ a)
35	30, 34	bitri 240	. . . . . . . . . . . . . . . . . . . 20 |- (<.a, <.b, <.c, d>.>.>. e. Ins2 Ins2 `' ~~ <-> d ~~ a)
36	29, 35	anbi12i 678	. . . . . . . . . . . . . . . . . . 19 |- ((<.a, <.b, <.c, d>.>.>. e. Ins4 `' Cross /\ <.a, <.b, <.c, d>.>.>. e. Ins2 Ins2 `' ~~ ) <-> (a = (b X. c) /\ d ~~ a))
37	23, 36	bitri 240	. . . . . . . . . . . . . . . . . 18 |- (<.a, <.b, <.c, d>.>.>. e. ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) <-> (a = (b X. c) /\ d ~~ a))
38	37	exbii 1582	. . . . . . . . . . . . . . . . 17 |- (E.a<.a, <.b, <.c, d>.>.>. e. ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) <-> E.a(a = (b X. c) /\ d ~~ a))
39	16, 7	xpex 5116	. . . . . . . . . . . . . . . . . 18 |- (b X. c) e. _V
40		breq2 4644	. . . . . . . . . . . . . . . . . 18 |- (a = (b X. c) -> (d ~~ a <-> d ~~ (b X. c)))
41	39, 40	ceqsexv 2895	. . . . . . . . . . . . . . . . 17 |- (E.a(a = (b X. c) /\ d ~~ a) <-> d ~~ (b X. c))
42	22, 38, 41	3bitri 262	. . . . . . . . . . . . . . . 16 |- (<.b, <.c, d>.>. e. ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) <-> d ~~ (b X. c))
43	19, 21, 42	3bitri 262	. . . . . . . . . . . . . . 15 |- (<.{b}, <.{c}, <.{d}, m>.>.>. e. Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) <-> d ~~ (b X. c))
44	18, 43	anbi12i 678	. . . . . . . . . . . . . 14 |- ((<.{b}, <.{c}, <.{d}, m>.>.>. e. Ins2 Ins2 S /\ <.{b}, <.{c}, <.{d}, m>.>.>. e. Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ )) <-> (b e. m /\ d ~~ (b X. c)))
45	12, 44	bitri 240	. . . . . . . . . . . . 13 |- (<.{b}, <.{c}, <.{d}, m>.>.>. e. ( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ )) <-> (b e. m /\ d ~~ (b X. c)))
46	45	exbii 1582	. . . . . . . . . . . 12 |- (E.b<.{b}, <.{c}, <.{d}, m>.>.>. e. ( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ )) <-> E.b(b e. m /\ d ~~ (b X. c)))
47		elima1c 4948	. . . . . . . . . . . 12 |- (<.{c}, <.{d}, m>.>. e. (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c) <-> E.b<.{b}, <.{c}, <.{d}, m>.>.>. e. ( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ )))
48		df-rex 2621	. . . . . . . . . . . 12 |- (E.b e. m d ~~ (b X. c) <-> E.b(b e. m /\ d ~~ (b X. c)))
49	46, 47, 48	3bitr4i 268	. . . . . . . . . . 11 |- (<.{c}, <.{d}, m>.>. e. (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c) <-> E.b e. m d ~~ (b X. c))
50	11, 49	bitri 240	. . . . . . . . . 10 |- (<.{c}, <.{d}, <.m, n>.>.>. e. Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c) <-> E.b e. m d ~~ (b X. c))
51	10, 50	anbi12i 678	. . . . . . . . 9 |- ((<.{c}, <.{d}, <.m, n>.>.>. e. Ins2 Ins2 S /\ <.{c}, <.{d}, <.m, n>.>.>. e. Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c)) <-> (c e. n /\ E.b e. m d ~~ (b X. c)))
52	2, 51	bitri 240	. . . . . . . 8 |- (<.{c}, <.{d}, <.m, n>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c)) <-> (c e. n /\ E.b e. m d ~~ (b X. c)))
53	52	exbii 1582	. . . . . . 7 |- (E.c<.{c}, <.{d}, <.m, n>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c)) <-> E.c(c e. n /\ E.b e. m d ~~ (b X. c)))
54		df-rex 2621	. . . . . . 7 |- (E.c e. n E.b e. m d ~~ (b X. c) <-> E.c(c e. n /\ E.b e. m d ~~ (b X. c)))
55	53, 54	bitr4i 243	. . . . . 6 |- (E.c<.{c}, <.{d}, <.m, n>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c)) <-> E.c e. n E.b e. m d ~~ (b X. c))
56		elima1c 4948	. . . . . 6 |- (<.{d}, <.m, n>.>. e. (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c))"1c) <-> E.c<.{c}, <.{d}, <.m, n>.>.>. e. ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c)))
57		rexcom 2773	. . . . . 6 |- (E.b e. m E.c e. n d ~~ (b X. c) <-> E.c e. n E.b e. m d ~~ (b X. c))
58	55, 56, 57	3bitr4i 268	. . . . 5 |- (<.{d}, <.m, n>.>. e. (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c))"1c) <-> E.b e. m E.c e. n d ~~ (b X. c))
59		breq1 4643	. . . . . . 7 |- (a = d -> (a ~~ (b X. c) <-> d ~~ (b X. c)))
60	59	2rexbidv 2658	. . . . . 6 |- (a = d -> (E.b e. m E.c e. n a ~~ (b X. c) <-> E.b e. m E.c e. n d ~~ (b X. c)))
61	20, 60	elab 2986	. . . . 5 |- (d e. {a | E.b e. m E.c e. n a ~~ (b X. c)} <-> E.b e. m E.c e. n d ~~ (b X. c))
62	58, 61	bitr4i 243	. . . 4 |- (<.{d}, <.m, n>.>. e. (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c))"1c) <-> d e. {a | E.b e. m E.c e. n a ~~ (b X. c)})
63	62	releqmpt2 5810	. . 3 |- ((( NC X. NC ) X. _V) \ (( Ins2 S (+) Ins3 (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c))"1c))"1c)) = (m e. NC , n e. NC |-> {a | E.b e. m E.c e. n a ~~ (b X. c)})
64	1, 63	eqtr4i 2376	. 2 |- ·c = ((( NC X. NC ) X. _V) \ (( Ins2 S (+) Ins3 (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c))"1c))"1c))
65		ncsex 6112	. . 3 |- NC e. _V
66		ssetex 4745	. . . . . . 7 |- S e. _V
67	66	ins2ex 5798	. . . . . 6 |- Ins2 S e. _V
68	67	ins2ex 5798	. . . . 5 |- Ins2 Ins2 S e. _V
69		crossex 5851	. . . . . . . . . . . . . 14 |- Cross e. _V
70	69	cnvex 5103	. . . . . . . . . . . . 13 |- `' Cross e. _V
71	70	ins4ex 5800	. . . . . . . . . . . 12 |- Ins4 `' Cross e. _V
72		enex 6032	. . . . . . . . . . . . . . 15 |- ~~ e. _V
73	72	cnvex 5103	. . . . . . . . . . . . . 14 |- `' ~~ e. _V
74	73	ins2ex 5798	. . . . . . . . . . . . 13 |- Ins2 `' ~~ e. _V
75	74	ins2ex 5798	. . . . . . . . . . . 12 |- Ins2 Ins2 `' ~~ e. _V
76	71, 75	inex 4106	. . . . . . . . . . 11 |- ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) e. _V
77	76	rnex 5108	. . . . . . . . . 10 |- ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) e. _V
78	77	si3ex 5807	. . . . . . . . 9 |- SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) e. _V
79	78	ins4ex 5800	. . . . . . . 8 |- Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ) e. _V
80	68, 79	inex 4106	. . . . . . 7 |- ( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ )) e. _V
81		1cex 4143	. . . . . . 7 |- 1c e. _V
82	80, 81	imaex 4748	. . . . . 6 |- (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c) e. _V
83	82	ins4ex 5800	. . . . 5 |- Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c) e. _V
84	68, 83	inex 4106	. . . 4 |- ( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c)) e. _V
85	84, 81	imaex 4748	. . 3 |- (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c))"1c) e. _V
86	65, 65, 85	mpt2exlem 5812	. 2 |- ((( NC X. NC ) X. _V) \ (( Ins2 S (+) Ins3 (( Ins2 Ins2 S i^i Ins4 (( Ins2 Ins2 S i^i Ins4 SI3 ran ( Ins4 `' Cross i^i Ins2 Ins2 `' ~~ ))"1c))"1c))"1c)) e. _V
87	64, 86	eqeltri 2423	1 |- ·c e. _V

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  E.wrex 2616  _Vcvv 2860   \ cdif 3207   i^i cin 3209   (+) csymdif 3210  {csn 3738  1_cc1c 4135  <.cop 4562   class class class wbr 4640   S csset 4720  "cima 4723   X. cxp 4771  `'ccnv 4772  ran crn 4774   |-> cmpt2 5654   Ins2 cins2 5750   Ins3 cins3 5752   Ins4 cins4 5756   SI₃ csi3 5758   Cross ccross 5764   ~~ cen 6029   NC cncs 6089   ·_c cmuc 6093

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-csb 3138  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-cross 5765  df-ec 5948  df-qs 5952  df-en 6030  df-ncs 6099  df-muc 6103

This theorem is referenced by: (None)