Theorem pprodss4v 31991

Description: The parallel product is a subclass of ( ( _V X. _V ) X. ( _V X. _V ) ). (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
pprodss4v	|- pprod ( A , B ) C_ ( ( _V X. _V ) X. ( _V X. _V ) )

Proof of Theorem pprodss4v

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

Step	Hyp	Ref	Expression
1		df-pprod 31962	. 2 |- pprod ( A , B ) = ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) )
2		txprel 31986	. . 3 |- Rel ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) )
3		txpss3v 31985	. . . . . . 7 |- ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) C_ ( _V X. ( _V X. _V ) )
4	3	sseli 3599	. . . . . 6 |- ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> <. x , y >. e. ( _V X. ( _V X. _V ) ) )
5		opelxp2 5151	. . . . . 6 |- ( <. x , y >. e. ( _V X. ( _V X. _V ) ) -> y e. ( _V X. _V ) )
6	4, 5	syl 17	. . . . 5 |- ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> y e. ( _V X. _V ) )
7		elvv 5177	. . . . . 6 |- ( y e. ( _V X. _V ) <-> E. z E. w y = <. z , w >. )
8		opeq2 4403	. . . . . . . . 9 |- ( y = <. z , w >. -> <. x , y >. = <. x , <. z , w >. >. )
9	8	eleq1d 2686	. . . . . . . 8 |- ( y = <. z , w >. -> ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) <-> <. x , <. z , w >. >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) ) )
10		df-br 4654	. . . . . . . . 9 |- ( x ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) <. z , w >. <-> <. x , <. z , w >. >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) )
11		vex 3203	. . . . . . . . . . 11 |- x e. _V
12		vex 3203	. . . . . . . . . . 11 |- z e. _V
13		vex 3203	. . . . . . . . . . 11 |- w e. _V
14	11, 12, 13	brtxp 31987	. . . . . . . . . 10 |- ( x ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) <. z , w >. <-> ( x ( A o. ( 1st |` ( _V X. _V ) ) ) z /\ x ( B o. ( 2nd |` ( _V X. _V ) ) ) w ) )
15	11, 12	brco 5292	. . . . . . . . . . . 12 |- ( x ( A o. ( 1st |` ( _V X. _V ) ) ) z <-> E. y ( x ( 1st |` ( _V X. _V ) ) y /\ y A z ) )
16		vex 3203	. . . . . . . . . . . . . . . 16 |- y e. _V
17	16	brres 5402	. . . . . . . . . . . . . . 15 |- ( x ( 1st |` ( _V X. _V ) ) y <-> ( x 1st y /\ x e. ( _V X. _V ) ) )
18	17	simprbi 480	. . . . . . . . . . . . . 14 |- ( x ( 1st |` ( _V X. _V ) ) y -> x e. ( _V X. _V ) )
19	18	adantr 481	. . . . . . . . . . . . 13 |- ( ( x ( 1st |` ( _V X. _V ) ) y /\ y A z ) -> x e. ( _V X. _V ) )
20	19	exlimiv 1858	. . . . . . . . . . . 12 |- ( E. y ( x ( 1st |` ( _V X. _V ) ) y /\ y A z ) -> x e. ( _V X. _V ) )
21	15, 20	sylbi 207	. . . . . . . . . . 11 |- ( x ( A o. ( 1st |` ( _V X. _V ) ) ) z -> x e. ( _V X. _V ) )
22	21	adantr 481	. . . . . . . . . 10 |- ( ( x ( A o. ( 1st |` ( _V X. _V ) ) ) z /\ x ( B o. ( 2nd |` ( _V X. _V ) ) ) w ) -> x e. ( _V X. _V ) )
23	14, 22	sylbi 207	. . . . . . . . 9 |- ( x ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) <. z , w >. -> x e. ( _V X. _V ) )
24	10, 23	sylbir 225	. . . . . . . 8 |- ( <. x , <. z , w >. >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> x e. ( _V X. _V ) )
25	9, 24	syl6bi 243	. . . . . . 7 |- ( y = <. z , w >. -> ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> x e. ( _V X. _V ) ) )
26	25	exlimivv 1860	. . . . . 6 |- ( E. z E. w y = <. z , w >. -> ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> x e. ( _V X. _V ) ) )
27	7, 26	sylbi 207	. . . . 5 |- ( y e. ( _V X. _V ) -> ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> x e. ( _V X. _V ) ) )
28	6, 27	mpcom 38	. . . 4 |- ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> x e. ( _V X. _V ) )
29		opelxp 5146	. . . 4 |- ( <. x , y >. e. ( ( _V X. _V ) X. ( _V X. _V ) ) <-> ( x e. ( _V X. _V ) /\ y e. ( _V X. _V ) ) )
30	28, 6, 29	sylanbrc 698	. . 3 |- ( <. x , y >. e. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) -> <. x , y >. e. ( ( _V X. _V ) X. ( _V X. _V ) ) )
31	2, 30	relssi 5211	. 2 |- ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) C_ ( ( _V X. _V ) X. ( _V X. _V ) )
32	1, 31	eqsstri 3635	1 |- pprod ( A , B ) C_ ( ( _V X. _V ) X. ( _V X. _V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   C_ wss 3574   <.cop 4183   class class class wbr 4653   X. cxp 5112   |` cres 5116   o. ccom 5118   1stc1st 7166   2ndc2nd 7167   (x) ctxp 31937  pprodcpprod 31938

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-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-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-fv 5896  df-1st 7168  df-2nd 7169  df-txp 31961  df-pprod 31962

This theorem is referenced by:  brpprod3a  31993