Theorem relopabVD 39137

Description: Virtual deduction proof of relopab 5247. The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel L. O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. relopab 5247 is relopabVD 39137 without virtual deductions and was automatically derived from relopabVD 39137.

1::	|- (. y = v ->. y = v ).
2:1:	|- (. y = v ->. <. x ,. y >. = <. x ,. v >. ).
3::	|- (. y = v ,. x = u ->. x = u ).
4:3:	|- (. y = v ,. x = u ->. <. x ,. v >. = <. u , v >. ).
5:2,4:	|- (. y = v ,. x = u ->. <. x ,. y >. = <. u , v >. ).
6:5:	|- (. y = v ,. x = u ->. ( z = <. x ,. y >. -> z = <. u , v >. ) ).
7:6:	|- (. y = v ->. ( x = u -> ( z = <. x ,. y >. -> z = <. u , v >. ) ) ).
8:7:	|- ( y = v -> ( x = u -> ( z = <. x ,. y >. -> z = <. u , v >. ) ) )
9:8:	|- ( E. v y = v -> E. v ( x = u -> ( z = <. x , y >. -> z = <. u , v >. ) ) )
90::	|- ( v = y <-> y = v )
91:90:	|- ( E. v v = y <-> E. v y = v )
92::	|- E. v v = y
10:91,92:	|- E. v y = v
11:9,10:	|- E. v ( x = u -> ( z = <. x ,. y >. -> z = <. u , v >. ) )
12:11:	|- ( x = u -> E. v ( z = <. x ,. y >. -> z = <. u , v >. ) )
13::	|- ( E. v ( z = <. x ,. y >. -> z = <. u , v >. ) -> ( z = <. x , y >. -> E. v z = <. u , v >. ) )
14:12,13:	|- ( x = u -> ( z = <. x ,. y >. -> E. v z = <. u , v >. ) )
15:14:	|- ( E. u x = u -> E. u ( z = <. x ,. y >. -> E. v z = <. u , v >. ) )
150::	|- ( u = x <-> x = u )
151:150:	|- ( E. u u = x <-> E. u x = u )
152::	|- E. u u = x
16:151,152:	|- E. u x = u
17:15,16:	|- E. u ( z = <. x ,. y >. -> E. v z = <. u , v >. )
18:17:	|- ( z = <. x ,. y >. -> E. u E. v z = <. u , v >. )
19:18:	|- ( E. y z = <. x ,. y >. -> E. y E. u E. v z = <. u , v >. )
20::	|- ( E. y E. u E. v z = <. u ,. v >. -> E. u E. v z = <. u , v >. )
21:19,20:	|- ( E. y z = <. x ,. y >. -> E. u E. v z = <. u , v >. )
22:21:	|- ( E. x E. y z = <. x ,. y >. -> E. x E. u E. v z = <. u , v >. )
23::	|- ( E. x E. u E. v z = <. u ,. v >. -> E. u E. v z = <. u , v >. )
24:22,23:	|- ( E. x E. y z = <. x ,. y >. -> E. u E. v z = <. u , v >. )
25:24:	|- { z | E. x E. y z = <. x ,. y >. } C_ { z | E. u E. v z = <. u , v >. }
26::	|- x e. _V
27::	|- y e. _V
28:26,27:	|- ( x e. _V /\ y e. _V )
29:28:	|- ( z = <. x ,. y >. <-> ( z = <. x ,. y >. /\ ( x e. _V /\ y e. _V ) ) )
30:29:	|- ( E. y z = <. x ,. y >. <-> E. y ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) )
31:30:	|- ( E. x E. y z = <. x ,. y >. <-> E. x E. y ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) )
32:31:	|- { z | E. x E. y z = <. x ,. y >. } = { z | E. x E. y ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) }
320:25,32:	|- { z | E. x E. y ( z = <. x ,. y >. /\ ( x e. _V /\ y e. _V ) ) } C_ { z | E. u E. v z = <. u , v >. }
33::	|- u e. _V
34::	|- v e. _V
35:33,34:	|- ( u e. _V /\ v e. _V )
36:35:	|- ( z = <. u ,. v >. <-> ( z = <. u ,. v >. /\ ( u e. _V /\ v e. _V ) ) )
37:36:	|- ( E. v z = <. u ,. v >. <-> E. v ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) )
38:37:	|- ( E. u E. v z = <. u ,. v >. <-> E. u E. v ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) )
39:38:	|- { z | E. u E. v z = <. u ,. v >. } = { z | E. u E. v ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) }
40:320,39:	|- { z | E. x E. y ( z = <. x ,. y >. /\ ( x e. _V /\ y e. _V ) ) } C_ { z | E. u E. v ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) }
41::	|- { <. x ,. y >. | ( x e. _V /\ y e. _V ) } = { z | E. x E. y ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) }
42::	|- { <. u ,. v >. | ( u e. _V /\ v e. _V ) } = { z | E. u E. v ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) }
43:40,41,42:	|- { <. x ,. y >. | ( x e. _V /\ y e. _V ) } C_ { <. u , v >. | ( u e. _V /\ v e. _V ) }
44::	|- { <. u ,. v >. | ( u e. _V /\ v e. _V ) } = ( _V X. _V )
45:43,44:	|- { <. x ,. y >. | ( x e. _V /\ y e. _V ) } C_ ( _V X. _V )
46:28:	|- ( ph -> ( x e. _V /\ y e. _V ) )
47:46:	|- { <. x ,. y >. | ph } C_ { <. x ,. y >. | ( x e. _V /\ y e. _V ) }
48:45,47:	|- { <. x ,. y >. | ph } C_ ( _V X. _V )
qed:48:	|- Rel { <. x ,. y >. | ph }

(Contributed by Alan Sare, 9-Jul-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
relopabVD	|- Rel { <. x , y >. | ph }

Proof of Theorem relopabVD

Dummy variables z v u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . 6 |- x e. _V
2		vex 3203	. . . . . 6 |- y e. _V
3	1, 2	pm3.2i 471	. . . . 5 |- ( x e. _V /\ y e. _V )
4	3	a1i 11	. . . 4 |- ( ph -> ( x e. _V /\ y e. _V ) )
5	4	ssopab2i 5003	. . 3 |- { <. x , y >. | ph } C_ { <. x , y >. | ( x e. _V /\ y e. _V ) }
6	3	biantru 526	. . . . . . . . . 10 |- ( z = <. x , y >. <-> ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) )
7	6	exbii 1774	. . . . . . . . 9 |- ( E. y z = <. x , y >. <-> E. y ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) )
8	7	exbii 1774	. . . . . . . 8 |- ( E. x E. y z = <. x , y >. <-> E. x E. y ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) )
9	8	abbii 2739	. . . . . . 7 |- { z | E. x E. y z = <. x , y >. } = { z | E. x E. y ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) }
10		ax6ev 1890	. . . . . . . . . . . . . . 15 |- E. u u = x
11		equcom 1945	. . . . . . . . . . . . . . . 16 |- ( u = x <-> x = u )
12	11	exbii 1774	. . . . . . . . . . . . . . 15 |- ( E. u u = x <-> E. u x = u )
13	10, 12	mpbi 220	. . . . . . . . . . . . . 14 |- E. u x = u
14		ax6ev 1890	. . . . . . . . . . . . . . . . . . 19 |- E. v v = y
15		equcom 1945	. . . . . . . . . . . . . . . . . . . 20 |- ( v = y <-> y = v )
16	15	exbii 1774	. . . . . . . . . . . . . . . . . . 19 |- ( E. v v = y <-> E. v y = v )
17	14, 16	mpbi 220	. . . . . . . . . . . . . . . . . 18 |- E. v y = v
18		idn1 38790	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (. y = v ->. y = v ).
19		opeq2 4403	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( y = v -> <. x , y >. = <. x , v >. )
20	18, 19	e1a 38852	. . . . . . . . . . . . . . . . . . . . . . 23 |- (. y = v ->. <. x , y >. = <. x , v >. ).
21		idn2 38838	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (. y = v ,. x = u ->. x = u ).
22		opeq1 4402	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( x = u -> <. x , v >. = <. u , v >. )
23	21, 22	e2 38856	. . . . . . . . . . . . . . . . . . . . . . 23 |- (. y = v ,. x = u ->. <. x , v >. = <. u , v >. ).
24		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( <. x , y >. = <. x , v >. -> ( <. x , y >. = <. u , v >. <-> <. x , v >. = <. u , v >. ) )
25	24	biimprd 238	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( <. x , y >. = <. x , v >. -> ( <. x , v >. = <. u , v >. -> <. x , y >. = <. u , v >. ) )
26	20, 23, 25	e12 38951	. . . . . . . . . . . . . . . . . . . . . 22 |- (. y = v ,. x = u ->. <. x , y >. = <. u , v >. ).
27		eqeq2 2633	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( <. x , y >. = <. u , v >. -> ( z = <. x , y >. <-> z = <. u , v >. ) )
28	27	biimpd 219	. . . . . . . . . . . . . . . . . . . . . 22 |- ( <. x , y >. = <. u , v >. -> ( z = <. x , y >. -> z = <. u , v >. ) )
29	26, 28	e2 38856	. . . . . . . . . . . . . . . . . . . . 21 |- (. y = v ,. x = u ->. ( z = <. x , y >. -> z = <. u , v >. ) ).
30	29	in2 38830	. . . . . . . . . . . . . . . . . . . 20 |- (. y = v ->. ( x = u -> ( z = <. x , y >. -> z = <. u , v >. ) ) ).
31	30	in1 38787	. . . . . . . . . . . . . . . . . . 19 |- ( y = v -> ( x = u -> ( z = <. x , y >. -> z = <. u , v >. ) ) )
32	31	eximi 1762	. . . . . . . . . . . . . . . . . 18 |- ( E. v y = v -> E. v ( x = u -> ( z = <. x , y >. -> z = <. u , v >. ) ) )
33	17, 32	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- E. v ( x = u -> ( z = <. x , y >. -> z = <. u , v >. ) )
34	33	19.37iv 1911	. . . . . . . . . . . . . . . 16 |- ( x = u -> E. v ( z = <. x , y >. -> z = <. u , v >. ) )
35		19.37v 1910	. . . . . . . . . . . . . . . . 17 |- ( E. v ( z = <. x , y >. -> z = <. u , v >. ) <-> ( z = <. x , y >. -> E. v z = <. u , v >. ) )
36	35	biimpi 206	. . . . . . . . . . . . . . . 16 |- ( E. v ( z = <. x , y >. -> z = <. u , v >. ) -> ( z = <. x , y >. -> E. v z = <. u , v >. ) )
37	34, 36	syl 17	. . . . . . . . . . . . . . 15 |- ( x = u -> ( z = <. x , y >. -> E. v z = <. u , v >. ) )
38	37	eximi 1762	. . . . . . . . . . . . . 14 |- ( E. u x = u -> E. u ( z = <. x , y >. -> E. v z = <. u , v >. ) )
39	13, 38	ax-mp 5	. . . . . . . . . . . . 13 |- E. u ( z = <. x , y >. -> E. v z = <. u , v >. )
40	39	19.37iv 1911	. . . . . . . . . . . 12 |- ( z = <. x , y >. -> E. u E. v z = <. u , v >. )
41	40	eximi 1762	. . . . . . . . . . 11 |- ( E. y z = <. x , y >. -> E. y E. u E. v z = <. u , v >. )
42		19.9v 1896	. . . . . . . . . . . 12 |- ( E. y E. u E. v z = <. u , v >. <-> E. u E. v z = <. u , v >. )
43	42	biimpi 206	. . . . . . . . . . 11 |- ( E. y E. u E. v z = <. u , v >. -> E. u E. v z = <. u , v >. )
44	41, 43	syl 17	. . . . . . . . . 10 |- ( E. y z = <. x , y >. -> E. u E. v z = <. u , v >. )
45	44	eximi 1762	. . . . . . . . 9 |- ( E. x E. y z = <. x , y >. -> E. x E. u E. v z = <. u , v >. )
46		19.9v 1896	. . . . . . . . . 10 |- ( E. x E. u E. v z = <. u , v >. <-> E. u E. v z = <. u , v >. )
47	46	biimpi 206	. . . . . . . . 9 |- ( E. x E. u E. v z = <. u , v >. -> E. u E. v z = <. u , v >. )
48	45, 47	syl 17	. . . . . . . 8 |- ( E. x E. y z = <. x , y >. -> E. u E. v z = <. u , v >. )
49	48	ss2abi 3674	. . . . . . 7 |- { z | E. x E. y z = <. x , y >. } C_ { z | E. u E. v z = <. u , v >. }
50	9, 49	eqsstr3i 3636	. . . . . 6 |- { z | E. x E. y ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) } C_ { z | E. u E. v z = <. u , v >. }
51		vex 3203	. . . . . . . . . . 11 |- u e. _V
52		vex 3203	. . . . . . . . . . 11 |- v e. _V
53	51, 52	pm3.2i 471	. . . . . . . . . 10 |- ( u e. _V /\ v e. _V )
54	53	biantru 526	. . . . . . . . 9 |- ( z = <. u , v >. <-> ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) )
55	54	exbii 1774	. . . . . . . 8 |- ( E. v z = <. u , v >. <-> E. v ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) )
56	55	exbii 1774	. . . . . . 7 |- ( E. u E. v z = <. u , v >. <-> E. u E. v ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) )
57	56	abbii 2739	. . . . . 6 |- { z | E. u E. v z = <. u , v >. } = { z | E. u E. v ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) }
58	50, 57	sseqtri 3637	. . . . 5 |- { z | E. x E. y ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) } C_ { z | E. u E. v ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) }
59		df-opab 4713	. . . . 5 |- { <. x , y >. | ( x e. _V /\ y e. _V ) } = { z | E. x E. y ( z = <. x , y >. /\ ( x e. _V /\ y e. _V ) ) }
60		df-opab 4713	. . . . 5 |- { <. u , v >. | ( u e. _V /\ v e. _V ) } = { z | E. u E. v ( z = <. u , v >. /\ ( u e. _V /\ v e. _V ) ) }
61	58, 59, 60	3sstr4i 3644	. . . 4 |- { <. x , y >. | ( x e. _V /\ y e. _V ) } C_ { <. u , v >. | ( u e. _V /\ v e. _V ) }
62		df-xp 5120	. . . . 5 |- ( _V X. _V ) = { <. u , v >. | ( u e. _V /\ v e. _V ) }
63	62	eqcomi 2631	. . . 4 |- { <. u , v >. | ( u e. _V /\ v e. _V ) } = ( _V X. _V )
64	61, 63	sseqtri 3637	. . 3 |- { <. x , y >. | ( x e. _V /\ y e. _V ) } C_ ( _V X. _V )
65	5, 64	sstri 3612	. 2 |- { <. x , y >. | ph } C_ ( _V X. _V )
66		df-rel 5121	. . 3 |- ( Rel { <. x , y >. | ph } <-> { <. x , y >. | ph } C_ ( _V X. _V ) )
67	66	biimpri 218	. 2 |- ( { <. x , y >. | ph } C_ ( _V X. _V ) -> Rel { <. x , y >. | ph } )
68	65, 67	e0a 38999	1 |- Rel { <. x , y >. | ph }

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   _Vcvv 3200   C_ wss 3574   <.cop 4183   {copab 4712   X. cxp 5112   Rel wrel 5119

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

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-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-vd1 38786  df-vd2 38794

This theorem is referenced by: (None)