Theorem eloprabi 4108

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 3983	. . . 4 |- Rel {<.<.x, y>., z>. | ph}
2		1st2nd 4098	. . . 4 |- ((Rel {<.<.x, y>., z>. | ph} /\ A e. {<.<.x, y>., z>. | ph}) -> A = <.(1st` A), (2nd` A)>.)
3	1, 2	mpan 694	. . 3 |- (A e. {<.<.x, y>., z>. | ph} -> A = <.(1st` A), (2nd` A)>.)
4		dfoprab3 4104	. . . . . 6 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | (w e. (V X. V) /\ [(1st` w) / x][(2nd` w) / y]ph)}
5	4	eleq2i 1535	. . . . 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 2818	. . . . . . 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 2061	. . . . . 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 1531	. . . . . . 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 4107	. . . . . 6 |- (A e. {<.w, z>. | w e. (V X. V)} -> (1st` A) e. (V X. V))
12		relxp 3250	. . . . . . 7 |- Rel (V X. V)
13		1st2nd 4098	. . . . . . 7 |- ((Rel (V X. V) /\ (1st` A) e. (V X. V)) -> (1st` A) = <.(1st` (1st` A)), (2nd` (1st` A))>.)
14	12, 13	mpan 694	. . . . . 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 2489	. . 3 |- (A e. {<.<.x, y>., z>. | ph} -> <.(1st` A), (2nd` A)>. = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>.)
18	3, 17	eqtrd 1504	. 2 |- (A e. {<.<.x, y>., z>. | ph} -> A = <.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>.)
19		eleq1 1531	. . . 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 3723	. . . . 5 |- (1st` (1st` A)) e. V
21		fvex 3723	. . . . 5 |- (2nd` (1st` A)) e. V
22		fvex 3723	. . . . 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 3998	. . . . 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 914	. . . 4 |- (<.<.(1st` (1st` A)), (2nd` (1st` A))>., (2nd` A)>. e. {<.<.x, y>., z>. | ph} <-> th)
28	19, 27	syl6bb 535	. . 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 954   e. wcel 956  [wsbc 1168  Vcvv 1807  <.cop 2407  {copab 2661   X. cxp 3163  Rel wrel 3170  ` cfv 3177  {copab2 3955  1stc1st 4067  2ndc2nd 4068

This theorem is referenced by:  nvi 8185

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193  df-oprab 3957  df-1st 4069  df-2nd 4070

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