Theorem aceq5lem3 4747

Description: Lemma for aceq5 4750.

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 2756	. . . 4 |- {w} e. V
2		visset 1816	. . . 4 |- w e. V
3	1, 2	xpex 3266	. . 3 |- ({w} X. w) e. V
4		neeq1 1593	. . . 4 |- (u = ({w} X. w) -> (u =/= (/) <-> ({w} X. w) =/= (/)))
5		eqeq1 1484	. . . . 5 |- (u = ({w} X. w) -> (u = ({t} X. t) <-> ({w} X. w) = ({t} X. t)))
6	5	rexbidv 1667	. . . 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 630	. . 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 1900	. 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 1541	. 2 |- (({w} X. w) e. A <-> ({w} X. w) e. {u | (u =/= (/) /\ E.t e. h u = ({t} X. t))})
11		xpeq2 3207	. . . . . 6 |- (w = (/) -> ({w} X. w) = ({w} X. (/)))
12		xp0 3471	. . . . . 6 |- ({w} X. (/)) = (/)
13	11, 12	syl6eq 1526	. . . . 5 |- (w = (/) -> ({w} X. w) = (/))
14		rneq 3345	. . . . . 6 |- (({w} X. w) = (/) -> ran ({w} X. w) = ran (/))
15	2	snnz 2462	. . . . . . 7 |- {w} =/= (/)
16		rnxp 3478	. . . . . . 7 |- ({w} =/= (/) -> ran ({w} X. w) = w)
17	15, 16	ax-mp 7	. . . . . 6 |- ran ({w} X. w) = w
18		rn0 3361	. . . . . 6 |- ran (/) = (/)
19	14, 17, 18	3eqtr3g 1533	. . . . 5 |- (({w} X. w) = (/) -> w = (/))
20	13, 19	impbi 157	. . . 4 |- (w = (/) <-> ({w} X. w) = (/))
21	20	necon3bii 1601	. . 3 |- (w =/= (/) <-> ({w} X. w) =/= (/))
22		df-rex 1653	. . . 4 |- (E.t e. h ({w} X. w) = ({t} X. t) <-> E.t(t e. h /\ ({w} X. w) = ({t} X. t)))
23		rneq 3345	. . . . . . . . . 10 |- (({w} X. w) = ({t} X. t) -> ran ({w} X. w) = ran ({t} X. t))
24		visset 1816	. . . . . . . . . . . 12 |- t e. V
25	24	snnz 2462	. . . . . . . . . . 11 |- {t} =/= (/)
26		rnxp 3478	. . . . . . . . . . 11 |- ({t} =/= (/) -> ran ({t} X. t) = t)
27	25, 26	ax-mp 7	. . . . . . . . . 10 |- ran ({t} X. t) = t
28	23, 17, 27	3eqtr3g 1533	. . . . . . . . 9 |- (({w} X. w) = ({t} X. t) -> w = t)
29		sneq 2421	. . . . . . . . . . 11 |- (w = t -> {w} = {t})
30		xpeq1 3206	. . . . . . . . . . 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 3207	. . . . . . . . . 10 |- (w = t -> ({t} X. w) = ({t} X. t))
33	31, 32	eqtrd 1510	. . . . . . . . 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 1480	. . . . . . . 8 |- (w = t <-> t = w)
36	34, 35	bitr 173	. . . . . . 7 |- (({w} X. w) = ({t} X. t) <-> t = w)
37	36	anbi2i 482	. . . . . 6 |- ((t e. h /\ ({w} X. w) = ({t} X. t)) <-> (t e. h /\ t = w))
38		ancom 437	. . . . . 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 1053	. . . 4 |- (E.t(t e. h /\ ({w} X. w) = ({t} X. t)) <-> E.t(t = w /\ t e. h))
41		eleq1 1537	. . . . 5 |- (t = w -> (t e. h <-> w e. h))
42	2, 41	ceqsexv 1838	. . . 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 484	. 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 958   e. wcel 960  E.wex 982  {cab 1466   =/= wne 1588  E.wrex 1649  (/)c0 2283  {csn 2413   X. cxp 3174  ran crn 3177

This theorem is referenced by:  aceq5lem5 4749

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195

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