Theorem eloprabi 4102

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 3977	. . . 4 |- Rel {<.<.x, y>., z>. | ph}
2		1st2nd 4092	. . . 4 |- ((Rel {<.<.x, y>., z>. | ph} /\ A e. {<.<.x, y>., z>. | ph}) -> A = <.(1st` A), (2nd` A)>.)
3	1, 2	mpan 693	. . 3 |- (A e. {<.<.x, y>., z>. | ph} -> A = <.(1st` A), (2nd` A)>.)
4		dfoprab3 4098	. . . . . 6 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)}
5	4	eleq2i 1530	. . . . 5 |- (A e. {<.<.x, y>., z>. | ph} <-> A e. {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)})
6		pm3.26 319	. . . . . . . 8 |- ((w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph) -> w e. (V X. V))
7	6	ssopab2i 2812	. . . . . . 7 |- {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)} (_ {<.w, z>. | w e. (V X. V)}
8	7	sseli 2055	. . . . . 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 1526	. . . . . . 7 |- (w = (1st` A) -> (w e. (V X. V) <-> (1st` A) e. (V X. V)))
10		pm4.2i 171	. . . . . . 7 |- (z = (2nd` A) -> ((1st` A) e. (V X. V) <-> (1st` A) e. (V X. V)))
11	9, 10	elopabi 4101	. . . . . 6 |- (A e. {<.w, z>. | w e. (V X. V)} -> (1st` A) e. (V X. V))
12		relxp 3245	. . . . . . 7 |- Rel (V X. V)
13		1st2nd 4092	. . . . . . 7 |- ((Rel (V X. V) /\ (1st` A) e. (V X. V)) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
14	12, 13	mpan 693	. . . . . 6 |- ((1st` A) e. (V X. V) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
15	8, 11, 14	3syl 20	. . . . 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 199	. . . 4 |- (A e. {<.<.x, y>., z>. | ph} -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
17	16	opeq1d 2484	. . 3 |- (A e. {<.<.x, y>., z>. | ph} -> <.(1st` A), (2nd` A)>. = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>.)
18	3, 17	eqtrd 1499	. 2 |- (A e. {<.<.x, y>., z>. | ph} -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>.)
19		eleq1 1526	. . . 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 3717	. . . . 5 |- (1st` (1st` A)) e. V
21		fvex 3717	. . . . 5 |- (2nd` (1st` A)) e. V
22		fvex 3717	. . . . 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 3992	. . . . 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 913	. . . 4 |- (<.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. e. {<.<.x, y>., z>. | ph} <-> th)
28	19, 27	syl6bb 534	. . 3 |- (A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. -> (A e. {<.<.x, y>., z>. | ph} <-> th))
29	28	biimpcd 155	. 2 |- (A e. {<.<.x, y>., z>. | ph} -> (A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. -> th))
30	18, 29	mpd 26	1 |- (A e. {<.<.x, y>., z>. | ph} -> th)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  [wsbc 1166  Vcvv 1802  <.cop 2401  {copab 2656   X. cxp 3158  Rel wrel 3165  ` cfv 3172  {copab2 3949  1stc1st 4061  2ndc2nd 4062

This theorem is referenced by:  nvi 8173

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-oprab 3951  df-1st 4063  df-2nd 4064

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