Theorem opabex3d 6229

Description: Existence of an ordered pair abstraction, deduction version. (Contributed by Alexander van der Vekens, 19-Oct-2017.)

Hypotheses

Ref	Expression
opabex3d.1	|- ( ph -> A e. _V )
opabex3d.2	|- ( ( ph /\ x e. A ) -> { y | ps } e. _V )

Assertion

Ref	Expression
opabex3d	|- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V )

Distinct variable groups:   x, A, y   ph, x

Allowed substitution hints:   ph( y)   ps( x, y)

Proof of Theorem opabex3d

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

Step	Hyp	Ref	Expression
1		19.42v 1931	. . . . . 6 |- ( E. y ( x e. A /\ ( z = <. x , y >. /\ ps ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ps ) ) )
2		an12 561	. . . . . . 7 |- ( ( z = <. x , y >. /\ ( x e. A /\ ps ) ) <-> ( x e. A /\ ( z = <. x , y >. /\ ps ) ) )
3	2	exbii 1629	. . . . . 6 |- ( E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) <-> E. y ( x e. A /\ ( z = <. x , y >. /\ ps ) ) )
4		elxp 4710	. . . . . . . 8 |- ( z e. ( { x } X. { y | ps } ) <-> E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) )
5		excom 1688	. . . . . . . . 9 |- ( E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> E. w E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) )
6		an12 561	. . . . . . . . . . . . 13 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
7		velsn 3660	. . . . . . . . . . . . . 14 |- ( v e. { x } <-> v = x )
8	7	anbi1i 458	. . . . . . . . . . . . 13 |- ( ( v e. { x } /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
9	6, 8	bitri 184	. . . . . . . . . . . 12 |- ( ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
10	9	exbii 1629	. . . . . . . . . . 11 |- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) )
11		vex 2779	. . . . . . . . . . . 12 |- x e. _V
12		opeq1 3833	. . . . . . . . . . . . . 14 |- ( v = x -> <. v , w >. = <. x , w >. )
13	12	eqeq2d 2219	. . . . . . . . . . . . 13 |- ( v = x -> ( z = <. v , w >. <-> z = <. x , w >. ) )
14	13	anbi1d 465	. . . . . . . . . . . 12 |- ( v = x -> ( ( z = <. v , w >. /\ w e. { y | ps } ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) ) )
15	11, 14	ceqsexv 2816	. . . . . . . . . . 11 |- ( E. v ( v = x /\ ( z = <. v , w >. /\ w e. { y | ps } ) ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) )
16	10, 15	bitri 184	. . . . . . . . . 10 |- ( E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> ( z = <. x , w >. /\ w e. { y | ps } ) )
17	16	exbii 1629	. . . . . . . . 9 |- ( E. w E. v ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> E. w ( z = <. x , w >. /\ w e. { y | ps } ) )
18	5, 17	bitri 184	. . . . . . . 8 |- ( E. v E. w ( z = <. v , w >. /\ ( v e. { x } /\ w e. { y | ps } ) ) <-> E. w ( z = <. x , w >. /\ w e. { y | ps } ) )
19		nfv 1552	. . . . . . . . . 10 |- F/ y z = <. x , w >.
20		nfsab1 2197	. . . . . . . . . 10 |- F/ y w e. { y | ps }
21	19, 20	nfan 1589	. . . . . . . . 9 |- F/ y ( z = <. x , w >. /\ w e. { y | ps } )
22		nfv 1552	. . . . . . . . 9 |- F/ w ( z = <. x , y >. /\ ps )
23		opeq2 3834	. . . . . . . . . . 11 |- ( w = y -> <. x , w >. = <. x , y >. )
24	23	eqeq2d 2219	. . . . . . . . . 10 |- ( w = y -> ( z = <. x , w >. <-> z = <. x , y >. ) )
25		df-clab 2194	. . . . . . . . . . 11 |- ( w e. { y | ps } <-> [ w / y ] ps )
26		sbequ12 1795	. . . . . . . . . . . 12 |- ( y = w -> ( ps <-> [ w / y ] ps ) )
27	26	equcoms 1732	. . . . . . . . . . 11 |- ( w = y -> ( ps <-> [ w / y ] ps ) )
28	25, 27	bitr4id 199	. . . . . . . . . 10 |- ( w = y -> ( w e. { y | ps } <-> ps ) )
29	24, 28	anbi12d 473	. . . . . . . . 9 |- ( w = y -> ( ( z = <. x , w >. /\ w e. { y | ps } ) <-> ( z = <. x , y >. /\ ps ) ) )
30	21, 22, 29	cbvex 1780	. . . . . . . 8 |- ( E. w ( z = <. x , w >. /\ w e. { y | ps } ) <-> E. y ( z = <. x , y >. /\ ps ) )
31	4, 18, 30	3bitri 206	. . . . . . 7 |- ( z e. ( { x } X. { y | ps } ) <-> E. y ( z = <. x , y >. /\ ps ) )
32	31	anbi2i 457	. . . . . 6 |- ( ( x e. A /\ z e. ( { x } X. { y | ps } ) ) <-> ( x e. A /\ E. y ( z = <. x , y >. /\ ps ) ) )
33	1, 3, 32	3bitr4ri 213	. . . . 5 |- ( ( x e. A /\ z e. ( { x } X. { y | ps } ) ) <-> E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) )
34	33	exbii 1629	. . . 4 |- ( E. x ( x e. A /\ z e. ( { x } X. { y | ps } ) ) <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) )
35		eliun 3945	. . . . 5 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> E. x e. A z e. ( { x } X. { y | ps } ) )
36		df-rex 2492	. . . . 5 |- ( E. x e. A z e. ( { x } X. { y | ps } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ps } ) ) )
37	35, 36	bitri 184	. . . 4 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> E. x ( x e. A /\ z e. ( { x } X. { y | ps } ) ) )
38		elopab 4322	. . . 4 |- ( z e. { <. x , y >. | ( x e. A /\ ps ) } <-> E. x E. y ( z = <. x , y >. /\ ( x e. A /\ ps ) ) )
39	34, 37, 38	3bitr4i 212	. . 3 |- ( z e. U_ x e. A ( { x } X. { y | ps } ) <-> z e. { <. x , y >. | ( x e. A /\ ps ) } )
40	39	eqriv 2204	. 2 |- U_ x e. A ( { x } X. { y | ps } ) = { <. x , y >. | ( x e. A /\ ps ) }
41		opabex3d.1	. . 3 |- ( ph -> A e. _V )
42		snexg 4244	. . . . . 6 |- ( x e. _V -> { x } e. _V )
43	11, 42	ax-mp 5	. . . . 5 |- { x } e. _V
44		opabex3d.2	. . . . 5 |- ( ( ph /\ x e. A ) -> { y | ps } e. _V )
45		xpexg 4807	. . . . 5 |- ( ( { x } e. _V /\ { y | ps } e. _V ) -> ( { x } X. { y | ps } ) e. _V )
46	43, 44, 45	sylancr 414	. . . 4 |- ( ( ph /\ x e. A ) -> ( { x } X. { y | ps } ) e. _V )
47	46	ralrimiva 2581	. . 3 |- ( ph -> A. x e. A ( { x } X. { y | ps } ) e. _V )
48		iunexg 6227	. . 3 |- ( ( A e. _V /\ A. x e. A ( { x } X. { y | ps } ) e. _V ) -> U_ x e. A ( { x } X. { y | ps } ) e. _V )
49	41, 47, 48	syl2anc 411	. 2 |- ( ph -> U_ x e. A ( { x } X. { y | ps } ) e. _V )
50	40, 49	eqeltrrid 2295	1 |- ( ph -> { <. x , y >. | ( x e. A /\ ps ) } e. _V )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   E.wex 1516   [wsb 1786   e. wcel 2178   {cab 2193   A.wral 2486   E.wrex 2487   _Vcvv 2776   {csn 3643   <.cop 3646   U_ciun 3941   {copab 4120   X. cxp 4691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298

This theorem is referenced by:  acfun  7350  ccfunen  7411  ovshftex  11245