Theorem opabn1stprc 7228

Description: An ordered-pair class abstraction which does not depend on the first abstraction variable is a proper class. There must be, however, at least one set which satisfies the restricting wwf. (Contributed by AV, 27-Dec-2020.)

Assertion

Ref	Expression
opabn1stprc	|- ( E. y ph -> { <. x , y >. | ph } e/ _V )

Distinct variable groups:   x, y   ph, x

Allowed substitution hint:   ph( y)

Proof of Theorem opabn1stprc

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . . 8 |- x e. _V
2	1	biantrur 527	. . . . . . 7 |- ( ph <-> ( x e. _V /\ ph ) )
3	2	opabbii 4717	. . . . . 6 |- { <. x , y >. | ph } = { <. x , y >. | ( x e. _V /\ ph ) }
4	3	dmeqi 5325	. . . . 5 |- dom { <. x , y >. | ph } = dom { <. x , y >. | ( x e. _V /\ ph ) }
5		id 22	. . . . . . 7 |- ( E. y ph -> E. y ph )
6	5	ralrimivw 2967	. . . . . 6 |- ( E. y ph -> A. x e. _V E. y ph )
7		dmopab3 5337	. . . . . 6 |- ( A. x e. _V E. y ph <-> dom { <. x , y >. | ( x e. _V /\ ph ) } = _V )
8	6, 7	sylib 208	. . . . 5 |- ( E. y ph -> dom { <. x , y >. | ( x e. _V /\ ph ) } = _V )
9	4, 8	syl5eq 2668	. . . 4 |- ( E. y ph -> dom { <. x , y >. | ph } = _V )
10		vprc 4796	. . . . 5 |- -. _V e. _V
11	10	a1i 11	. . . 4 |- ( E. y ph -> -. _V e. _V )
12	9, 11	eqneltrd 2720	. . 3 |- ( E. y ph -> -. dom { <. x , y >. | ph } e. _V )
13		dmexg 7097	. . 3 |- ( { <. x , y >. | ph } e. _V -> dom { <. x , y >. | ph } e. _V )
14	12, 13	nsyl 135	. 2 |- ( E. y ph -> -. { <. x , y >. | ph } e. _V )
15		df-nel 2898	. 2 |- ( { <. x , y >. | ph } e/ _V <-> -. { <. x , y >. | ph } e. _V )
16	14, 15	sylibr 224	1 |- ( E. y ph -> { <. x , y >. | ph } e/ _V )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   e/ wnel 2897   A.wral 2912   _Vcvv 3200   {copab 4712   dom cdm 5114

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-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-nel 2898  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-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125

This theorem is referenced by:  griedg0prc  26156