Theorem aceq5lem1 4659

Description: Lemma for aceq5 4664.

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:   w,v,y,g

Proof of Theorem aceq5lem1

Step	Hyp	Ref	Expression
1		elin 2178	. . . 4 |- (v e. (({w} X. w) i^i y) <-> (v e. ({w} X. w) /\ v e. y))
2		elxp 3165	. . . . . 6 |- (v e. ({w} X. w) <-> E.tE.g(v = <.t, g>. /\ (t e. {w} /\ g e. w)))
3		excom 1022	. . . . . 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 173	. . . . 5 |- (v e. ({w} X. w) <-> E.gE.t(v = <.t, g>. /\ (t e. {w} /\ g e. w)))
5	4	anbi1i 480	. . . 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 1288	. . . . 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 484	. . . . . . . . 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 1510	. . . . . . . . . . 11 |- (v = <.t, g>. -> (v e. y <-> <.t, g>. e. y))
9	8	pm5.32i 643	. . . . . . . . . 10 |- ((v = <.t, g>. /\ v e. y) <-> (v = <.t, g>. /\ <.t, g>. e. y))
10		elsn 2392	. . . . . . . . . . 11 |- (t e. {w} <-> t = w)
11	10	anbi1i 480	. . . . . . . . . 10 |- ((t e. {w} /\ g e. w) <-> (t = w /\ g e. w))
12	9, 11	anbi12i 481	. . . . . . . . 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 505	. . . . . . . . . 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 435	. . . . . . . . . . 11 |- ((v = <.t, g>. /\ t = w) <-> (t = w /\ v = <.t, g>.))
15		ancom 435	. . . . . . . . . . 11 |- ((<.t, g>. e. y /\ g e. w) <-> (g e. w /\ <.t, g>. e. y))
16	14, 15	anbi12i 481	. . . . . . . . . 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 439	. . . . . . . . . 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 177	. . . . . . . . 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 177	. . . . . . . 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	exbii 1027	. . . . . . 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 1788	. . . . . . . 8 |- w e. V
22		opeq1 2456	. . . . . . . . . 10 |- (t = w -> <.t, g>. = <.w, g>.)
23	22	eqeq2d 1462	. . . . . . . . 9 |- (t = w -> (v = <.t, g>. <-> v = <.w, g>.))
24	22	eleq1d 1516	. . . . . . . . . 10 |- (t = w -> (<.t, g>. e. y <-> <.w, g>. e. y))
25	24	anbi2d 614	. . . . . . . . 9 |- (t = w -> ((g e. w /\ <.t, g>. e. y) <-> (g e. w /\ <.w, g>. e. y)))
26	23, 25	anbi12d 626	. . . . . . . 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 1810	. . . . . . 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 173	. . . . . 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	exbii 1027	. . . . 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 175	. . . 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 177	. . 3 |- (v e. (({w} X. w) i^i y) <-> E.g(v = <.w, g>. /\ (g e. w /\ <.w, g>. e. y)))
32	31	eubii 1364	. 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 2769	. 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 173	1 |- (E!v v e. (({w} X. w) i^i y) <-> E!g(g e. w /\ <.w, g>. e. y))

Syntax hints:   <-> wb 146   /\ wa 223  E.wex 956   = wceq 1099   e. wcel 1105  E!weu 1357   i^i cin 2017  {csn 2380  <.cop 2382   X. cxp 3131

This theorem is referenced by:  aceq5lem5 4663

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-opab 2635  df-xp 3147

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