Theorem eloprabi 4176

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 4050	. . . 4 |- Rel {<.<.x, y>., z>. | ph}
2		1st2nd 4166	. . . 4 |- ((Rel {<.<.x, y>., z>. | ph} /\ A e. {<.<.x, y>., z>. | ph}) -> A = <.(1st` A), (2nd` A)>.)
3	1, 2	mpan 707	. . 3 |- (A e. {<.<.x, y>., z>. | ph} -> A = <.(1st` A), (2nd` A)>.)
4		dfoprab3 4172	. . . . . 6 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)}
5	4	eleq2i 1585	. . . . 5 |- (A e. {<.<.x, y>., z>. | ph} <-> A e. {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)})
6		pm3.26 326	. . . . . . . 8 |- ((w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph) -> w e. (V X. V))
7	6	ssopab2i 2879	. . . . . . 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 2116	. . . . . 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 1581	. . . . . . 7 |- (w = (1st` A) -> (w e. (V X. V) <-> (1st` A) e. (V X. V)))
10		pm4.2d 178	. . . . . . 7 |- (z = (2nd` A) -> ((1st` A) e. (V X. V) <-> (1st` A) e. (V X. V)))
11	9, 10	elopabi 4175	. . . . . 6 |- (A e. {<.w, z>. | w e. (V X. V)} -> (1st` A) e. (V X. V))
12		relxp 3312	. . . . . . 7 |- Rel (V X. V)
13		1st2nd 4166	. . . . . . 7 |- ((Rel (V X. V) /\ (1st` A) e. (V X. V)) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
14	12, 13	mpan 707	. . . . . 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 206	. . . 4 |- (A e. {<.<.x, y>., z>. | ph} -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
17	16	opeq1d 2547	. . 3 |- (A e. {<.<.x, y>., z>. | ph} -> <.(1st` A), (2nd` A)>. = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>.)
18	3, 17	eqtrd 1554	. 2 |- (A e. {<.<.x, y>., z>. | ph} -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>.)
19		eleq1 1581	. . . 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 3789	. . . . 5 |- (1st` (1st` A)) e. V
21		fvex 3789	. . . . 5 |- (2nd` (1st` A)) e. V
22		fvex 3789	. . . . 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 4065	. . . . 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 928	. . . 4 |- (<.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. e. {<.<.x, y>., z>. | ph} <-> th)
28	19, 27	syl6bb 547	. . 3 |- (A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. -> (A e. {<.<.x, y>., z>. | ph} <-> th))
29	28	biimpcd 162	. 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 153   /\ wa 230   = wceq 997   e. wcel 999  [wsbc 1212  Vcvv 1858  <.cop 2463  {copab 2721   X. cxp 3225  Rel wrel 3232  ` cfv 3239  {copab2 4022  1stc1st 4135  2ndc2nd 4136

This theorem is referenced by:  nvi 8317

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-9 1006  ax-10 1007  ax-11 1008  ax-12 1009  ax-13 1010  ax-14 1011  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182  ax-16 1252  ax-11o 1260  ax-ext 1504  ax-sep 2758  ax-nul 2765  ax-pow 2798  ax-pr 2835  ax-un 2922

This theorem depends on definitions:  df-bi 154  df-or 231  df-an 232  df-3an 789  df-ex 1022  df-sb 1214  df-eu 1424  df-mo 1425  df-clab 1510  df-cleq 1515  df-clel 1518  df-ne 1634  df-ral 1696  df-rex 1697  df-v 1859  df-sbc 1989  df-dif 2100  df-un 2101  df-in 2102  df-ss 2104  df-nul 2332  df-pw 2454  df-sn 2464  df-pr 2465  df-op 2468  df-uni 2558  df-br 2675  df-opab 2722  df-id 2891  df-xp 3241  df-rel 3242  df-cnv 3243  df-co 3244  df-dm 3245  df-rn 3246  df-res 3247  df-ima 3248  df-fun 3249  df-fv 3255  df-oprab 4024  df-1st 4137  df-2nd 4138

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