Theorem aceq5lem1 3558

Description: Lemma for aceq5 3563.

Assertion

Ref	Expression
aceq5lem1	|- (E!v v e. (({w} X. w) i^i y) <-> E!g(g e. w /\ <.w, g>. e. y))

Distinct variable group(s):   w,v,y,g

Proof of Theorem aceq5lem1

Step	Hyp	Ref	Expression
1		elin 1635	. . . 4 |- (v e. (({w} X. w) i^i y) <-> (v e. ({w} X. w) /\ v e. y))
2		elxp 2442	. . . . . 6 |- (v e. ({w} X. w) <-> E.tE.g(v = <.t, g>. /\ (t e. {w} /\ g e. w)))
3		excom 728	. . . . . 6 |- (E.tE.g(v = <.t, g>. /\ (t e. {w} /\ g e. w)) <-> E.gE.t(v = <.t, g>. /\ (t e. {w} /\ g e. w)))
4	2, 3	bitr 151	. . . . 5 |- (v e. ({w} X. w) <-> E.gE.t(v = <.t, g>. /\ (t e. {w} /\ g e. w)))
5	4	anbi1i 368	. . . 4 |- ((v e. ({w} X. w) /\ v e. y) <-> (E.gE.t(v = <.t, g>. /\ (t e. {w} /\ g e. w)) /\ v e. y))
6		19.41vv 964	. . . . 5 |- (E.gE.t((v = <.t, g>. /\ (t e. {w} /\ g e. w)) /\ v e. y) <-> (E.gE.t(v = <.t, g>. /\ (t e. {w} /\ g e. w)) /\ v e. y))
7		an23 371	. . . . . . . . 9 |- (((v = <.t, g>. /\ (t e. {w} /\ g e. w)) /\ v e. y) <-> ((v = <.t, g>. /\ v e. y) /\ (t e. {w} /\ g e. w)))
8		eleq1 1149	. . . . . . . . . . 11 |- (v = <.t, g>. -> (v e. y <-> <.t, g>. e. y))
9	8	pm5.32i 489	. . . . . . . . . 10 |- ((v = <.t, g>. /\ v e. y) <-> (v = <.t, g>. /\ <.t, g>. e. y))
10		elsn 1820	. . . . . . . . . . 11 |- (t e. {w} <-> t = w)
11	10	anbi1i 368	. . . . . . . . . 10 |- ((t e. {w} /\ g e. w) <-> (t = w /\ g e. w))
12	9, 11	anbi12i 369	. . . . . . . . 9 |- (((v = <.t, g>. /\ v e. y) /\ (t e. {w} /\ g e. w)) <-> ((v = <.t, g>. /\ <.t, g>. e. y) /\ (t = w /\ g e. w)))
13		an4 388	. . . . . . . . . 10 |- (((v = <.t, g>. /\ <.t, g>. e. y) /\ (t = w /\ g e. w)) <-> ((v = <.t, g>. /\ t = w) /\ (<.t, g>. e. y /\ g e. w)))
14		ancom 333	. . . . . . . . . . 11 |- ((v = <.t, g>. /\ t = w) <-> (t = w /\ v = <.t, g>.))
15		ancom 333	. . . . . . . . . . 11 |- ((<.t, g>. e. y /\ g e. w) <-> (g e. w /\ <.t, g>. e. y))
16	14, 15	anbi12i 369	. . . . . . . . . 10 |- (((v = <.t, g>. /\ t = w) /\ (<.t, g>. e. y /\ g e. w)) <-> ((t = w /\ v = <.t, g>.) /\ (g e. w /\ <.t, g>. e. y)))
17		anass 336	. . . . . . . . . 10 |- (((t = w /\ v = <.t, g>.) /\ (g e. w /\ <.t, g>. e. y)) <-> (t = w /\ (v = <.t, g>. /\ (g e. w /\ <.t, g>. e. y))))
18	13, 16, 17	3bitr 155	. . . . . . . . 9 |- (((v = <.t, g>. /\ <.t, g>. e. y) /\ (t = w /\ g e. w)) <-> (t = w /\ (v = <.t, g>. /\ (g e. w /\ <.t, g>. e. y))))
19	7, 12, 18	3bitr 155	. . . . . . . 8 |- (((v = <.t, g>. /\ (t e. {w} /\ g e. w)) /\ v e. y) <-> (t = w /\ (v = <.t, g>. /\ (g e. w /\ <.t, g>. e. y))))
20	19	biex 733	. . . . . . 7 |- (E.t((v = <.t, g>. /\ (t e. {w} /\ g e. w)) /\ v e. y) <-> E.t(t = w /\ (v = <.t, g>. /\ (g e. w /\ <.t, g>. e. y))))
21		visset 1350	. . . . . . . 8 |- w e. V
22		opeq1 1876	. . . . . . . . . 10 |- (t = w -> <.t, g>. = <.w, g>.)
23	22	cleq2d 1112	. . . . . . . . 9 |- (t = w -> (v = <.t, g>. <-> v = <.w, g>.))
24	22	eleq1d 1155	. . . . . . . . . 10 |- (t = w -> (<.t, g>. e. y <-> <.w, g>. e. y))
25	24	anbi2d 468	. . . . . . . . 9 |- (t = w -> ((g e. w /\ <.t, g>. e. y) <-> (g e. w /\ <.w, g>. e. y)))
26	23, 25	anbi12d 476	. . . . . . . 8 |- (t = w -> ((v = <.t, g>. /\ (g e. w /\ <.t, g>. e. y)) <-> (v = <.w, g>. /\ (g e. w /\ <.w, g>. e. y))))
27	21, 26	ceqsexv 1371	. . . . . . 7 |- (E.t(t = w /\ (v = <.t, g>. /\ (g e. w /\ <.t, g>. e. y))) <-> (v = <.w, g>. /\ (g e. w /\ <.w, g>. e. y)))
28	20, 27	bitr 151	. . . . . 6 |- (E.t((v = <.t, g>. /\ (t e. {w} /\ g e. w)) /\ v e. y) <-> (v = <.w, g>. /\ (g e. w /\ <.w, g>. e. y)))
29	28	biex 733	. . . . 5 |- (E.gE.t((v = <.t, g>. /\ (t e. {w} /\ g e. w)) /\ v e. y) <-> E.g(v = <.w, g>. /\ (g e. w /\ <.w, g>. e. y)))
30	6, 29	bitr3 153	. . . 4 |- ((E.gE.t(v = <.t, g>. /\ (t e. {w} /\ g e. w)) /\ v e. y) <-> E.g(v = <.w, g>. /\ (g e. w /\ <.w, g>. e. y)))
31	1, 5, 30	3bitr 155	. . 3 |- (v e. (({w} X. w) i^i y) <-> E.g(v = <.w, g>. /\ (g e. w /\ <.w, g>. e. y)))
32	31	bieu 1014	. 2 |- (E!v v e. (({w} X. w) i^i y) <-> E!vE.g(v = <.w, g>. /\ (g e. w /\ <.w, g>. e. y)))
33		euop2 1912	. 2 |- (E!vE.g(v = <.w, g>. /\ (g e. w /\ <.w, g>. e. y)) <-> E!g(g e. w /\ <.w, g>. e. y))
34	32, 33	bitr 151	1 |- (E!v v e. (({w} X. w) i^i y) <-> E!g(g e. w /\ <.w, g>. e. y))

Syntax hints:   <-> wb 127   /\ wa 196  E.wex 678   = weq 797   e. wel 803  E!weu 1007   = wceq 1091   e. wcel 1092   i^i cin 1486  {csn 1808  <.cop 1810   X. cxp 2408

This theorem is referenced by:  aceq5lem5 3562

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424

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