Theorem dfac5lem2 8947

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
dfac5lem2	|- ( <. w , g >. e. U. A <-> ( w e. h /\ g e. w ) )

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

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

Proof of Theorem dfac5lem2

Step	Hyp	Ref	Expression
1		dfac5lem.1	. . . 4 |- A = { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) }
2	1	unieqi 4445	. . 3 |- U. A = U. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) }
3	2	eleq2i 2693	. 2 |- ( <. w , g >. e. U. A <-> <. w , g >. e. U. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) } )
4		eluniab 4447	. . 3 |- ( <. w , g >. e. U. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) } <-> E. u ( <. w , g >. e. u /\ ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) ) )
5		r19.42v 3092	. . . . 5 |- ( E. t e. h ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> ( ( <. w , g >. e. u /\ u =/= (/) ) /\ E. t e. h u = ( { t } X. t ) ) )
6		anass 681	. . . . 5 |- ( ( ( <. w , g >. e. u /\ u =/= (/) ) /\ E. t e. h u = ( { t } X. t ) ) <-> ( <. w , g >. e. u /\ ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) ) )
7	5, 6	bitr2i 265	. . . 4 |- ( ( <. w , g >. e. u /\ ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) ) <-> E. t e. h ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) )
8	7	exbii 1774	. . 3 |- ( E. u ( <. w , g >. e. u /\ ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) ) <-> E. u E. t e. h ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) )
9		rexcom4 3225	. . . 4 |- ( E. t e. h E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> E. u E. t e. h ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) )
10		df-rex 2918	. . . 4 |- ( E. t e. h E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> E. t ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) )
11	9, 10	bitr3i 266	. . 3 |- ( E. u E. t e. h ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> E. t ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) )
12	4, 8, 11	3bitri 286	. 2 |- ( <. w , g >. e. U. { u | ( u =/= (/) /\ E. t e. h u = ( { t } X. t ) ) } <-> E. t ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) )
13		ancom 466	. . . . . . . . 9 |- ( ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> ( u = ( { t } X. t ) /\ ( <. w , g >. e. u /\ u =/= (/) ) ) )
14		ne0i 3921	. . . . . . . . . . 11 |- ( <. w , g >. e. u -> u =/= (/) )
15	14	pm4.71i 664	. . . . . . . . . 10 |- ( <. w , g >. e. u <-> ( <. w , g >. e. u /\ u =/= (/) ) )
16	15	anbi2i 730	. . . . . . . . 9 |- ( ( u = ( { t } X. t ) /\ <. w , g >. e. u ) <-> ( u = ( { t } X. t ) /\ ( <. w , g >. e. u /\ u =/= (/) ) ) )
17	13, 16	bitr4i 267	. . . . . . . 8 |- ( ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> ( u = ( { t } X. t ) /\ <. w , g >. e. u ) )
18	17	exbii 1774	. . . . . . 7 |- ( E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> E. u ( u = ( { t } X. t ) /\ <. w , g >. e. u ) )
19		snex 4908	. . . . . . . . 9 |- { t } e. _V
20		vex 3203	. . . . . . . . 9 |- t e. _V
21	19, 20	xpex 6962	. . . . . . . 8 |- ( { t } X. t ) e. _V
22		eleq2 2690	. . . . . . . 8 |- ( u = ( { t } X. t ) -> ( <. w , g >. e. u <-> <. w , g >. e. ( { t } X. t ) ) )
23	21, 22	ceqsexv 3242	. . . . . . 7 |- ( E. u ( u = ( { t } X. t ) /\ <. w , g >. e. u ) <-> <. w , g >. e. ( { t } X. t ) )
24	18, 23	bitri 264	. . . . . 6 |- ( E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) <-> <. w , g >. e. ( { t } X. t ) )
25	24	anbi2i 730	. . . . 5 |- ( ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) <-> ( t e. h /\ <. w , g >. e. ( { t } X. t ) ) )
26		opelxp 5146	. . . . . . 7 |- ( <. w , g >. e. ( { t } X. t ) <-> ( w e. { t } /\ g e. t ) )
27		velsn 4193	. . . . . . . . 9 |- ( w e. { t } <-> w = t )
28		equcom 1945	. . . . . . . . 9 |- ( w = t <-> t = w )
29	27, 28	bitri 264	. . . . . . . 8 |- ( w e. { t } <-> t = w )
30	29	anbi1i 731	. . . . . . 7 |- ( ( w e. { t } /\ g e. t ) <-> ( t = w /\ g e. t ) )
31	26, 30	bitri 264	. . . . . 6 |- ( <. w , g >. e. ( { t } X. t ) <-> ( t = w /\ g e. t ) )
32	31	anbi2i 730	. . . . 5 |- ( ( t e. h /\ <. w , g >. e. ( { t } X. t ) ) <-> ( t e. h /\ ( t = w /\ g e. t ) ) )
33		an12 838	. . . . 5 |- ( ( t e. h /\ ( t = w /\ g e. t ) ) <-> ( t = w /\ ( t e. h /\ g e. t ) ) )
34	25, 32, 33	3bitri 286	. . . 4 |- ( ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) <-> ( t = w /\ ( t e. h /\ g e. t ) ) )
35	34	exbii 1774	. . 3 |- ( E. t ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) <-> E. t ( t = w /\ ( t e. h /\ g e. t ) ) )
36		vex 3203	. . . 4 |- w e. _V
37		elequ1 1997	. . . . 5 |- ( t = w -> ( t e. h <-> w e. h ) )
38		eleq2 2690	. . . . 5 |- ( t = w -> ( g e. t <-> g e. w ) )
39	37, 38	anbi12d 747	. . . 4 |- ( t = w -> ( ( t e. h /\ g e. t ) <-> ( w e. h /\ g e. w ) ) )
40	36, 39	ceqsexv 3242	. . 3 |- ( E. t ( t = w /\ ( t e. h /\ g e. t ) ) <-> ( w e. h /\ g e. w ) )
41	35, 40	bitri 264	. 2 |- ( E. t ( t e. h /\ E. u ( ( <. w , g >. e. u /\ u =/= (/) ) /\ u = ( { t } X. t ) ) ) <-> ( w e. h /\ g e. w ) )
42	3, 12, 41	3bitri 286	1 |- ( <. w , g >. e. U. A <-> ( w e. h /\ g e. w ) )

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   <.cop 4183   U.cuni 4436   X. cxp 5112

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-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-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  dfac5lem5  8950