Theorem eloprabi 5060

Description: A consequence of membership in an operation class abstraction, using ordered pair extractors.

Hypotheses

Ref	Expression
eloprabi.1	|- (x = (1st` (1st` A)) -> (ph <-> ps))
eloprabi.2	|- (y = (2nd` (1st` A)) -> (ps <-> ch))
eloprabi.3	|- (z = (2nd` A) -> (ch <-> th))

Assertion

Ref	Expression
eloprabi	|- (A e. {<.<.x, y>., z>. | ph} -> th)

Distinct variable groups:   x,y,z,A   ch,x,y   ps,x   th,x,y,z

Proof of Theorem eloprabi

Step	Hyp	Ref	Expression
1		reloprab 4918	. . . 4 |- Rel {<.<.x, y>., z>. | ph}
2		1st2nd 5048	. . . 4 |- ((Rel {<.<.x, y>., z>. | ph} /\ A e. {<.<.x, y>., z>. | ph}) -> A = <.(1st` A), (2nd` A)>.)
3	1, 2	mpan 759	. . 3 |- (A e. {<.<.x, y>., z>. | ph} -> A = <.(1st` A), (2nd` A)>.)
4		dfoprab3 5054	. . . . . 6 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | (w e. (_V X. _V) /\ [(1st` w) / x][(2nd` w) / y]ph)}
5	4	eleq2i 1961	. . . . 5 |- (A e. {<.<.x, y>., z>. | ph} <-> A e. {<.w, z>. | (w e. (_V X. _V) /\ [(1st` w) / x][(2nd` w) / y]ph)})
6		simpl 346	. . . . . . . 8 |- ((w e. (_V X. _V) /\ [(1st` w) / x][(2nd` w) / y]ph) -> w e. (_V X. _V))
7	6	ssopab2i 3574	. . . . . . 7 |- {<.w, z>. | (w e. (_V X. _V) /\ [(1st` w) / x][(2nd` w) / y]ph)} C_ {<.w, z>. | w e. (_V X. _V)}
8	7	sseli 2617	. . . . . 6 |- (A e. {<.w, z>. | (w e. (_V X. _V) /\ [(1st` w) / x][(2nd` w) / y]ph)} -> A e. {<.w, z>. | w e. (_V X. _V)})
9		eleq1 1957	. . . . . . 7 |- (w = (1st` A) -> (w e. (_V X. _V) <-> (1st` A) e. (_V X. _V)))
10		biidd 188	. . . . . . 7 |- (z = (2nd` A) -> ((1st` A) e. (_V X. _V) <-> (1st` A) e. (_V X. _V)))
11	9, 10	elopabi 5059	. . . . . 6 |- (A e. {<.w, z>. | w e. (_V X. _V)} -> (1st` A) e. (_V X. _V))
12		relxp 4088	. . . . . . 7 |- Rel (_V X. _V)
13		1st2nd 5048	. . . . . . 7 |- ((Rel (_V X. _V) /\ (1st` A) e. (_V X. _V)) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
14	12, 13	mpan 759	. . . . . 6 |- ((1st` A) e. (_V X. _V) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
15	8, 11, 14	3syl 24	. . . . 5 |- (A e. {<.w, z>. | (w e. (_V X. _V) /\ [(1st` w) / x][(2nd` w) / y]ph)} -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
16	5, 15	sylbi 216	. . . 4 |- (A e. {<.<.x, y>., z>. | ph} -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
17	16	opeq1d 3164	. . 3 |- (A e. {<.<.x, y>., z>. | ph} -> <.(1st` A), (2nd` A)>. = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>.)
18	3, 17	eqtrd 1925	. 2 |- (A e. {<.<.x, y>., z>. | ph} -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>.)
19		eleq1 1957	. . . 4 |- (A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. -> (A e. {<.<.x, y>., z>. | ph} <-> <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. e. {<.<.x, y>., z>. | ph}))
20		fvex 4689	. . . . 5 |- (1st` (1st` A)) e. _V
21		fvex 4689	. . . . 5 |- (2nd` (1st` A)) e. _V
22		fvex 4689	. . . . 5 |- (2nd` A) e. _V
23		eloprabi.1	. . . . . 6 |- (x = (1st` (1st` A)) -> (ph <-> ps))
24		eloprabi.2	. . . . . 6 |- (y = (2nd` (1st` A)) -> (ps <-> ch))
25		eloprabi.3	. . . . . 6 |- (z = (2nd` A) -> (ch <-> th))
26	23, 24, 25	eloprabg 4936	. . . . 5 |- (((1st` (1st` A)) e. _V /\ (2nd` (1st` A)) e. _V /\ (2nd` A) e. _V) -> (<.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. e. {<.<.x, y>., z>. | ph} <-> th))
27	20, 21, 22, 26	mp3an 1191	. . . 4 |- (<.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. e. {<.<.x, y>., z>. | ph} <-> th)
28	19, 27	syl6bb 595	. . 3 |- (A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. -> (A e. {<.<.x, y>., z>. | ph} <-> th))
29	28	biimpcd 172	. 2 |- (A e. {<.<.x, y>., z>. | ph} -> (A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. -> th))
30	18, 29	mpd 29	1 |- (A e. {<.<.x, y>., z>. | ph} -> th)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  _Vcvv 2292  <.cop 3046  {copab 3395   X. cxp 3984  Rel wrel 3991  ` cfv 3998  {copab2 4885  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  nvi 9565

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-sbc 2454  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-oprab 4887  df-1st 5020  df-2nd 5021

Copyright terms: Public domain