Theorem elpglem3 42456

Description: Lemma for elpg 42457. (Contributed by Emmett Weisz, 28-Aug-2021.)

Assertion

Ref	Expression
elpglem3	|- ( E. x ( x C_ Pg /\ A e. ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) ) <-> ( A e. ( _V X. _V ) /\ E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )

Distinct variable groups:   x, A   x, y

Allowed substitution hint:   A( y)

Proof of Theorem elpglem3

Step	Hyp	Ref	Expression
1		vex 3203	. . . . . . . 8 |- x e. _V
2		pweq 4161	. . . . . . . . . 10 |- ( y = x -> ~P y = ~P x )
3	2	sqxpeqd 5141	. . . . . . . . 9 |- ( y = x -> ( ~P y X. ~P y ) = ( ~P x X. ~P x ) )
4		eqid 2622	. . . . . . . . 9 |- ( y e. _V |-> ( ~P y X. ~P y ) ) = ( y e. _V |-> ( ~P y X. ~P y ) )
5	1	pwex 4848	. . . . . . . . . 10 |- ~P x e. _V
6	5, 5	xpex 6962	. . . . . . . . 9 |- ( ~P x X. ~P x ) e. _V
7	3, 4, 6	fvmpt 6282	. . . . . . . 8 |- ( x e. _V -> ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) = ( ~P x X. ~P x ) )
8	1, 7	ax-mp 5	. . . . . . 7 |- ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) = ( ~P x X. ~P x )
9	8	eleq2i 2693	. . . . . 6 |- ( A e. ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) <-> A e. ( ~P x X. ~P x ) )
10		elxp7 7201	. . . . . 6 |- ( A e. ( ~P x X. ~P x ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) )
11	9, 10	bitri 264	. . . . 5 |- ( A e. ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) <-> ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) )
12	11	anbi2i 730	. . . 4 |- ( ( x C_ Pg /\ A e. ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) ) <-> ( x C_ Pg /\ ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )
13		an12 838	. . . 4 |- ( ( x C_ Pg /\ ( A e. ( _V X. _V ) /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) <-> ( A e. ( _V X. _V ) /\ ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )
14	12, 13	bitri 264	. . 3 |- ( ( x C_ Pg /\ A e. ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) ) <-> ( A e. ( _V X. _V ) /\ ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )
15	14	exbii 1774	. 2 |- ( E. x ( x C_ Pg /\ A e. ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) ) <-> E. x ( A e. ( _V X. _V ) /\ ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )
16		19.42v 1918	. 2 |- ( E. x ( A e. ( _V X. _V ) /\ ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) <-> ( A e. ( _V X. _V ) /\ E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )
17	15, 16	bitri 264	1 |- ( E. x ( x C_ Pg /\ A e. ( ( y e. _V |-> ( ~P y X. ~P y ) ) ` x ) ) <-> ( A e. ( _V X. _V ) /\ E. x ( x C_ Pg /\ ( ( 1st ` A ) e. ~P x /\ ( 2nd ` A ) e. ~P x ) ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   X. cxp 5112   ` cfv 5888   1stc1st 7166   2ndc2nd 7167  Pgcpg 42452

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169

This theorem is referenced by:  elpg  42457