Theorem opabn1stprc 6423

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 wff. (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 2824	. . . . . . . 8 |- x e. _V
2	1	biantrur 303	. . . . . . 7 |- ( ph <-> ( x e. _V /\ ph ) )
3	2	opabbii 4196	. . . . . 6 |- { <. x , y >. | ph } = { <. x , y >. | ( x e. _V /\ ph ) }
4	3	dmeqi 4980	. . . . 5 |- dom { <. x , y >. | ph } = dom { <. x , y >. | ( x e. _V /\ ph ) }
5		id 19	. . . . . . 7 |- ( E. y ph -> E. y ph )
6	5	ralrimivw 2624	. . . . . 6 |- ( E. y ph -> A. x e. _V E. y ph )
7		dmopab3 4992	. . . . . 6 |- ( A. x e. _V E. y ph <-> dom { <. x , y >. | ( x e. _V /\ ph ) } = _V )
8	6, 7	sylib 122	. . . . 5 |- ( E. y ph -> dom { <. x , y >. | ( x e. _V /\ ph ) } = _V )
9	4, 8	eqtrid 2283	. . . 4 |- ( E. y ph -> dom { <. x , y >. | ph } = _V )
10		vprc 4263	. . . . 5 |- -. _V e. _V
11	10	a1i 9	. . . 4 |- ( E. y ph -> -. _V e. _V )
12	9, 11	eqneltrd 2334	. . 3 |- ( E. y ph -> -. dom { <. x , y >. | ph } e. _V )
13		dmexg 5044	. . 3 |- ( { <. x , y >. | ph } e. _V -> dom { <. x , y >. | ph } e. _V )
14	12, 13	nsyl 637	. 2 |- ( E. y ph -> -. { <. x , y >. | ph } e. _V )
15		df-nel 2516	. 2 |- ( { <. x , y >. | ph } e/ _V <-> -. { <. x , y >. | ph } e. _V )
16	14, 15	sylibr 134	1 |- ( E. y ph -> { <. x , y >. | ph } e/ _V )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   E.wex 1545   e. wcel 2209   e/ wnel 2515   A.wral 2528   _Vcvv 2821   {copab 4189   dom cdm 4772

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-nel 2516  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-opab 4191  df-cnv 4780  df-dm 4782  df-rn 4783

This theorem is referenced by:  griedg0prc  16474