Theorem dfac5lem3 8948

Description: Lemma for dfac5 8951. (Contributed by NM, 12-Apr-2004.)

Hypothesis

Ref	Expression
dfac5lem.1	|- A = { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) }

Assertion

Ref	Expression
dfac5lem3	|- ( ( { w } X. w ) e. A <-> ( w =/= (/) /\ w e. h ) )

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

Allowed substitution hints:   A( u, t, h)

Proof of Theorem dfac5lem3

Step	Hyp	Ref	Expression
1		snex 4908	. . . 4 |- { w } e. _V
2		vex 3203	. . . 4 |- w e. _V
3	1, 2	xpex 6962	. . 3 |- ( { w } X. w ) e. _V
4		neeq1 2856	. . . 4 |- ( u = ( { w } X. w ) -> ( u =/= (/) <-> ( { w } X. w ) =/= (/) ) )
5		eqeq1 2626	. . . . 5 |- ( u = ( { w } X. w ) -> ( u = ( { t } X. t ) <-> ( { w } X. w ) = ( { t } X. t ) ) )
6	5	rexbidv 3052	. . . 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 747	. . 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 3350	. 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		dfac5lem.1	. . 3 |- A = { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) }
10	9	eleq2i 2693	. 2 |- ( ( { w } X. w ) e. A <-> ( { w } X. w ) e. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) } )
11		xpeq2 5129	. . . . . 6 |- ( w = (/) -> ( { w } X. w ) = ( { w } X. (/) ) )
12		xp0 5552	. . . . . 6 |- ( { w } X. (/) ) = (/)
13	11, 12	syl6eq 2672	. . . . 5 |- ( w = (/) -> ( { w } X. w ) = (/) )
14		rneq 5351	. . . . . 6 |- ( ( { w } X. w ) = (/) -> ran ( { w } X. w ) = ran (/) )
15	2	snnz 4309	. . . . . . 7 |- { w } =/= (/)
16		rnxp 5564	. . . . . . 7 |- ( { w } =/= (/) -> ran ( { w } X. w ) = w )
17	15, 16	ax-mp 5	. . . . . 6 |- ran ( { w } X. w ) = w
18		rn0 5377	. . . . . 6 |- ran (/) = (/)
19	14, 17, 18	3eqtr3g 2679	. . . . 5 |- ( ( { w } X. w ) = (/) -> w = (/) )
20	13, 19	impbii 199	. . . 4 |- ( w = (/) <-> ( { w } X. w ) = (/) )
21	20	necon3bii 2846	. . 3 |- ( w =/= (/) <-> ( { w } X. w ) =/= (/) )
22		df-rex 2918	. . . 4 |- ( E. t e. h ( { w } X. w ) = ( { t } X. t ) <-> E. t ( t e. h /\ ( { w } X. w ) = ( { t } X. t ) ) )
23		rneq 5351	. . . . . . . . . 10 |- ( ( { w } X. w ) = ( { t } X. t ) -> ran ( { w } X. w ) = ran ( { t } X. t ) )
24		vex 3203	. . . . . . . . . . . 12 |- t e. _V
25	24	snnz 4309	. . . . . . . . . . 11 |- { t } =/= (/)
26		rnxp 5564	. . . . . . . . . . 11 |- ( { t } =/= (/) -> ran ( { t } X. t ) = t )
27	25, 26	ax-mp 5	. . . . . . . . . 10 |- ran ( { t } X. t ) = t
28	23, 17, 27	3eqtr3g 2679	. . . . . . . . 9 |- ( ( { w } X. w ) = ( { t } X. t ) -> w = t )
29		sneq 4187	. . . . . . . . . . 11 |- ( w = t -> { w } = { t } )
30	29	xpeq1d 5138	. . . . . . . . . 10 |- ( w = t -> ( { w } X. w ) = ( { t } X. w ) )
31		xpeq2 5129	. . . . . . . . . 10 |- ( w = t -> ( { t } X. w ) = ( { t } X. t ) )
32	30, 31	eqtrd 2656	. . . . . . . . 9 |- ( w = t -> ( { w } X. w ) = ( { t } X. t ) )
33	28, 32	impbii 199	. . . . . . . 8 |- ( ( { w } X. w ) = ( { t } X. t ) <-> w = t )
34		equcom 1945	. . . . . . . 8 |- ( w = t <-> t = w )
35	33, 34	bitri 264	. . . . . . 7 |- ( ( { w } X. w ) = ( { t } X. t ) <-> t = w )
36	35	anbi2i 730	. . . . . 6 |- ( ( t e. h /\ ( { w } X. w ) = ( { t } X. t ) ) <-> ( t e. h /\ t = w ) )
37		ancom 466	. . . . . 6 |- ( ( t e. h /\ t = w ) <-> ( t = w /\ t e. h ) )
38	36, 37	bitri 264	. . . . 5 |- ( ( t e. h /\ ( { w } X. w ) = ( { t } X. t ) ) <-> ( t = w /\ t e. h ) )
39	38	exbii 1774	. . . 4 |- ( E. t ( t e. h /\ ( { w } X. w ) = ( { t } X. t ) ) <-> E. t ( t = w /\ t e. h ) )
40		elequ1 1997	. . . . 5 |- ( t = w -> ( t e. h <-> w e. h ) )
41	2, 40	ceqsexv 3242	. . . 4 |- ( E. t ( t = w /\ t e. h ) <-> w e. h )
42	22, 39, 41	3bitrri 287	. . 3 |- ( w e. h <-> E. t e. h ( { w } X. w ) = ( { t } X. t ) )
43	21, 42	anbi12i 733	. 2 |- ( ( w =/= (/) /\ w e. h ) <-> ( ( { w } X. w ) =/= (/) /\ E. t e. h ( { w } X. w ) = ( { t } X. t ) ) )
44	8, 10, 43	3bitr4i 292	1 |- ( ( { w } X. w ) e. A <-> ( w =/= (/) /\ w e. h ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   =/= wne 2794   E.wrex 2913   (/)c0 3915   {csn 4177   X. cxp 5112   ran crn 5115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  dfac5lem5  8950