Theorem aceq5lem3 4717

Description: Lemma for aceq5 4720.

Hypothesis

Ref	Expression
aceq5lem.1	|- A = {u | (u =/= (/) /\ E.t e. h u = ({t} X. t))}

Assertion

Ref	Expression
aceq5lem3	|- (({w} X. w) e. A <-> (w =/= (/) /\ w e. h))

Distinct variable groups:   w,u,t,h   w,A

Proof of Theorem aceq5lem3

Step	Hyp	Ref	Expression
1		snex 2745	. . . 4 |- {w} e. V
2		visset 1809	. . . 4 |- w e. V
3	1, 2	xpex 3255	. . 3 |- ({w} X. w) e. V
4		neeq1 1587	. . . 4 |- (u = ({w} X. w) -> (u =/= (/) <-> ({w} X. w) =/= (/)))
5		eqeq1 1478	. . . . 5 |- (u = ({w} X. w) -> (u = ({t} X. t) <-> ({w} X. w) = ({t} X. t)))
6	5	rexbidv 1661	. . . 4 |- (u = ({w} X. w) -> (E.t e. h u = ({t} X. t) <-> E.t e. h ({w} X. w) = ({t} X. t)))
7	4, 6	anbi12d 627	. . 3 |- (u = ({w} X. w) -> ((u =/= (/) /\ E.t e. h u = ({t} X. t)) <-> (({w} X. w) =/= (/) /\ E.t e. h ({w} X. w) = ({t} X. t))))
8	3, 7	elab 1893	. 2 |- (({w} X. w) e. {u | (u =/= (/) /\ E.t e. h u = ({t} X. t))} <-> (({w} X. w) =/= (/) /\ E.t e. h ({w} X. w) = ({t} X. t)))
9		aceq5lem.1	. . 3 |- A = {u | (u =/= (/) /\ E.t e. h u = ({t} X. t))}
10	9	eleq2i 1535	. 2 |- (({w} X. w) e. A <-> ({w} X. w) e. {u | (u =/= (/) /\ E.t e. h u = ({t} X. t))})
11		xpeq2 3196	. . . . . 6 |- (w = (/) -> ({w} X. w) = ({w} X. (/)))
12		xp0 3457	. . . . . 6 |- ({w} X. (/)) = (/)
13	11, 12	syl6eq 1520	. . . . 5 |- (w = (/) -> ({w} X. w) = (/))
14		rneq 3334	. . . . . 6 |- (({w} X. w) = (/) -> ran ({w} X. w) = ran (/))
15	2	snnz 2454	. . . . . . 7 |- {w} =/= (/)
16		rnxp 3464	. . . . . . 7 |- ({w} =/= (/) -> ran ({w} X. w) = w)
17	15, 16	ax-mp 7	. . . . . 6 |- ran ({w} X. w) = w
18		rn0 3349	. . . . . 6 |- ran (/) = (/)
19	14, 17, 18	3eqtr3g 1527	. . . . 5 |- (({w} X. w) = (/) -> w = (/))
20	13, 19	impbi 157	. . . 4 |- (w = (/) <-> ({w} X. w) = (/))
21	20	necon3bii 1595	. . 3 |- (w =/= (/) <-> ({w} X. w) =/= (/))
22		df-rex 1647	. . . 4 |- (E.t e. h ({w} X. w) = ({t} X. t) <-> E.t(t e. h /\ ({w} X. w) = ({t} X. t)))
23		rneq 3334	. . . . . . . . . 10 |- (({w} X. w) = ({t} X. t) -> ran ({w} X. w) = ran ({t} X. t))
24		visset 1809	. . . . . . . . . . . 12 |- t e. V
25	24	snnz 2454	. . . . . . . . . . 11 |- {t} =/= (/)
26		rnxp 3464	. . . . . . . . . . 11 |- ({t} =/= (/) -> ran ({t} X. t) = t)
27	25, 26	ax-mp 7	. . . . . . . . . 10 |- ran ({t} X. t) = t
28	23, 17, 27	3eqtr3g 1527	. . . . . . . . 9 |- (({w} X. w) = ({t} X. t) -> w = t)
29		sneq 2413	. . . . . . . . . . 11 |- (w = t -> {w} = {t})
30		xpeq1 3195	. . . . . . . . . . 11 |- ({w} = {t} -> ({w} X. w) = ({t} X. w))
31	29, 30	syl 10	. . . . . . . . . 10 |- (w = t -> ({w} X. w) = ({t} X. w))
32		xpeq2 3196	. . . . . . . . . 10 |- (w = t -> ({t} X. w) = ({t} X. t))
33	31, 32	eqtrd 1504	. . . . . . . . 9 |- (w = t -> ({w} X. w) = ({t} X. t))
34	28, 33	impbi 157	. . . . . . . 8 |- (({w} X. w) = ({t} X. t) <-> w = t)
35		eqcom 1474	. . . . . . . 8 |- (w = t <-> t = w)
36	34, 35	bitr 173	. . . . . . 7 |- (({w} X. w) = ({t} X. t) <-> t = w)
37	36	anbi2i 480	. . . . . 6 |- ((t e. h /\ ({w} X. w) = ({t} X. t)) <-> (t e. h /\ t = w))
38		ancom 435	. . . . . 6 |- ((t e. h /\ t = w) <-> (t = w /\ t e. h))
39	37, 38	bitr 173	. . . . 5 |- ((t e. h /\ ({w} X. w) = ({t} X. t)) <-> (t = w /\ t e. h))
40	39	exbii 1049	. . . 4 |- (E.t(t e. h /\ ({w} X. w) = ({t} X. t)) <-> E.t(t = w /\ t e. h))
41		eleq1 1531	. . . . 5 |- (t = w -> (t e. h <-> w e. h))
42	2, 41	ceqsexv 1831	. . . 4 |- (E.t(t = w /\ t e. h) <-> w e. h)
43	22, 40, 42	3bitrr 178	. . 3 |- (w e. h <-> E.t e. h ({w} X. w) = ({t} X. t))
44	21, 43	anbi12i 482	. 2 |- ((w =/= (/) /\ w e. h) <-> (({w} X. w) =/= (/) /\ E.t e. h ({w} X. w) = ({t} X. t)))
45	8, 10, 44	3bitr4 183	1 |- (({w} X. w) e. A <-> (w =/= (/) /\ w e. h))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978  {cab 1461   =/= wne 1582  E.wrex 1643  (/)c0 2276  {csn 2405   X. cxp 3163  ran crn 3166

This theorem is referenced by:  aceq5lem5 4719

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-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  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-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-xp 3179  df-rel 3180  df-cnv 3181  df-dm 3183  df-rn 3184

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