Theorem eloi 14400

Description: A consequence of membership in a class abstraction whose elements belong to ((_V X. _V) X. (_V X. _V)) using ordered pair extractors. (Used by category theory).

Hypotheses

Ref	Expression
eloi.1	|- (y = (1st` (1st` A)) -> (ph <-> ps))
eloi.2	|- (z = (2nd` (1st` A)) -> (ps <-> ch))
eloi.3	|- (v = (1st` (2nd` A)) -> (ch <-> th))
eloi.4	|- (w = (2nd` (2nd` A)) -> (th <-> ta))

Assertion

Ref	Expression
eloi	|- (A e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} -> ta)

Distinct variable groups:   v,A,w,x,y,z   ph,x   ta,v,w,y,z

Proof of Theorem eloi

Step	Hyp	Ref	Expression
1		eqeq1 1890	. . . . . . 7 |- (x = A -> (x = <.<.y, z>., <.v, w>.>. <-> A = <.<.y, z>., <.v, w>.>.))
2	1	anbi1d 679	. . . . . 6 |- (x = A -> ((x = <.<.y, z>., <.v, w>.>. /\ ph) <-> (A = <.<.y, z>., <.v, w>.>. /\ ph)))
3	2	4exbidv 1661	. . . . 5 |- (x = A -> (E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph) <-> E.yE.zE.vE.w(A = <.<.y, z>., <.v, w>.>. /\ ph)))
4	3	elabg 2405	. . . 4 |- (A e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} -> (A e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} <-> E.yE.zE.vE.w(A = <.<.y, z>., <.v, w>.>. /\ ph)))
5	4	ibi 652	. . 3 |- (A e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} -> E.yE.zE.vE.w(A = <.<.y, z>., <.v, w>.>. /\ ph))
6		id 73	. . . . . . . . 9 |- (A = <.<.y, z>., <.v, w>.>. -> A = <.<.y, z>., <.v, w>.>.)
7		opex 3527	. . . . . . . . . 10 |- <.y, z>. e. _V
8		opex 3527	. . . . . . . . . 10 |- <.v, w>. e. _V
9		opelxpi 4040	. . . . . . . . . 10 |- ((<.y, z>. e. _V /\ <.v, w>. e. _V) -> <.<.y, z>., <.v, w>.>. e. (_V X. _V))
10	7, 8, 9	mp2an 761	. . . . . . . . 9 |- <.<.y, z>., <.v, w>.>. e. (_V X. _V)
11	6, 10	syl6eqel 1979	. . . . . . . 8 |- (A = <.<.y, z>., <.v, w>.>. -> A e. (_V X. _V))
12		relxp 4088	. . . . . . . . 9 |- Rel (_V X. _V)
13		1st2nd 5048	. . . . . . . . 9 |- ((Rel (_V X. _V) /\ A e. (_V X. _V)) -> A = <.(1st` A), (2nd` A)>.)
14	12, 13	mpan 759	. . . . . . . 8 |- (A e. (_V X. _V) -> A = <.(1st` A), (2nd` A)>.)
15	11, 14	syl 12	. . . . . . 7 |- (A = <.<.y, z>., <.v, w>.>. -> A = <.(1st` A), (2nd` A)>.)
16		fveq2 4681	. . . . . . . . . . 11 |- (A = <.<.y, z>., <.v, w>.>. -> (1st` A) = (1st` <.<.y, z>., <.v, w>.>.))
17	7	op1st 5026	. . . . . . . . . . 11 |- (1st` <.<.y, z>., <.v, w>.>.) = <.y, z>.
18	16, 17	syl6eq 1944	. . . . . . . . . 10 |- (A = <.<.y, z>., <.v, w>.>. -> (1st` A) = <.y, z>.)
19		visset 2295	. . . . . . . . . . 11 |- y e. _V
20		visset 2295	. . . . . . . . . . 11 |- z e. _V
21		opelxpi 4040	. . . . . . . . . . 11 |- ((y e. _V /\ z e. _V) -> <.y, z>. e. (_V X. _V))
22	19, 20, 21	mp2an 761	. . . . . . . . . 10 |- <.y, z>. e. (_V X. _V)
23	18, 22	syl6eqel 1979	. . . . . . . . 9 |- (A = <.<.y, z>., <.v, w>.>. -> (1st` A) e. (_V X. _V))
24		1st2nd 5048	. . . . . . . . . 10 |- ((Rel (_V X. _V) /\ (1st` A) e. (_V X. _V)) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
25	12, 24	mpan 759	. . . . . . . . 9 |- ((1st` A) e. (_V X. _V) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
26	23, 25	syl 12	. . . . . . . 8 |- (A = <.<.y, z>., <.v, w>.>. -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
27		fveq2 4681	. . . . . . . . . . 11 |- (A = <.<.y, z>., <.v, w>.>. -> (2nd` A) = (2nd` <.<.y, z>., <.v, w>.>.))
28	7, 8	op2nd 5027	. . . . . . . . . . 11 |- (2nd` <.<.y, z>., <.v, w>.>.) = <.v, w>.
29	27, 28	syl6eq 1944	. . . . . . . . . 10 |- (A = <.<.y, z>., <.v, w>.>. -> (2nd` A) = <.v, w>.)
30		visset 2295	. . . . . . . . . . 11 |- v e. _V
31		visset 2295	. . . . . . . . . . 11 |- w e. _V
32		opelxpi 4040	. . . . . . . . . . 11 |- ((v e. _V /\ w e. _V) -> <.v, w>. e. (_V X. _V))
33	30, 31, 32	mp2an 761	. . . . . . . . . 10 |- <.v, w>. e. (_V X. _V)
34	29, 33	syl6eqel 1979	. . . . . . . . 9 |- (A = <.<.y, z>., <.v, w>.>. -> (2nd` A) e. (_V X. _V))
35		1st2nd 5048	. . . . . . . . . 10 |- ((Rel (_V X. _V) /\ (2nd` A) e. (_V X. _V)) -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.)
36	12, 35	mpan 759	. . . . . . . . 9 |- ((2nd` A) e. (_V X. _V) -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.)
37	34, 36	syl 12	. . . . . . . 8 |- (A = <.<.y, z>., <.v, w>.>. -> (2nd` A) = <.(1st` (2nd` A)), (2nd` (2nd` A))>.)
38	26, 37	opeq12d 3166	. . . . . . 7 |- (A = <.<.y, z>., <.v, w>.>. -> <.(1st` A), (2nd` A)>. = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>.)
39	15, 38	eqtrd 1925	. . . . . 6 |- (A = <.<.y, z>., <.v, w>.>. -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>.)
40	39	adantr 425	. . . . 5 |- ((A = <.<.y, z>., <.v, w>.>. /\ ph) -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>.)
41	40	19.23aivv 1675	. . . 4 |- (E.vE.w(A = <.<.y, z>., <.v, w>.>. /\ ph) -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>.)
42	41	19.23aivv 1675	. . 3 |- (E.yE.zE.vE.w(A = <.<.y, z>., <.v, w>.>. /\ ph) -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>.)
43	5, 42	syl 12	. 2 |- (A e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>.)
44		eleq1 1957	. . . 4 |- (A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>. -> (A e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} <-> <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>. e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)}))
45		fvex 4689	. . . . . 6 |- (1st` (1st` A)) e. _V
46		fvex 4689	. . . . . 6 |- (2nd` (1st` A)) e. _V
47		fvex 4689	. . . . . 6 |- (1st` (2nd` A)) e. _V
48	45, 46, 47	3pm3.2i 1048	. . . . 5 |- ((1st` (1st` A)) e. _V /\ (2nd` (1st` A)) e. _V /\ (1st` (2nd` A)) e. _V)
49		fvex 4689	. . . . 5 |- (2nd` (2nd` A)) e. _V
50		eloi.1	. . . . . 6 |- (y = (1st` (1st` A)) -> (ph <-> ps))
51		eloi.2	. . . . . 6 |- (z = (2nd` (1st` A)) -> (ps <-> ch))
52		eloi.3	. . . . . 6 |- (v = (1st` (2nd` A)) -> (ch <-> th))
53		eloi.4	. . . . . 6 |- (w = (2nd` (2nd` A)) -> (th <-> ta))
54	50, 51, 52, 53	elo 14342	. . . . 5 |- ((((1st` (1st` A)) e. _V /\ (2nd` (1st` A)) e. _V /\ (1st` (2nd` A)) e. _V) /\ (2nd` (2nd` A)) e. _V) -> (<.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>. e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} <-> ta))
55	48, 49, 54	mp2an 761	. . . 4 |- (<.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>. e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} <-> ta)
56	44, 55	syl6bb 595	. . 3 |- (A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>. -> (A e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} <-> ta))
57	56	biimpcd 172	. 2 |- (A e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} -> (A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., <.(1st` (2nd` A)), (2nd` (2nd` A))>.>. -> ta))
58	43, 57	mpd 29	1 |- (A e. {x | E.yE.zE.vE.w(x = <.<.y, z>., <.v, w>.>. /\ ph)} -> ta)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  _Vcvv 2292  <.cop 3046   X. cxp 3984  Rel wrel 3991  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  vecval3b 14795  algi 15074  dedi 15084  cati 15102

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-1st 5020  df-2nd 5021

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