Theorem brpprod 31992

Description: Characterize a quaternary relation over a tail Cartesian product. Together with pprodss4v 31991, this completely defines membership in a parallel product. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brpprod.1	|- X e. _V
brpprod.2	|- Y e. _V
brpprod.3	|- Z e. _V
brpprod.4	|- W e. _V

Assertion

Ref	Expression
brpprod	|- ( <. X , Y >.pprod ( A , B ) <. Z , W >. <-> ( X A Z /\ Y B W ) )

Proof of Theorem brpprod

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

Step	Hyp	Ref	Expression
1		df-pprod 31962	. . 3 |- pprod ( A , B ) = ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) )
2	1	breqi 4659	. 2 |- ( <. X , Y >.pprod ( A , B ) <. Z , W >. <-> <. X , Y >. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) <. Z , W >. )
3		opex 4932	. . 3 |- <. X , Y >. e. _V
4		brpprod.3	. . 3 |- Z e. _V
5		brpprod.4	. . 3 |- W e. _V
6	3, 4, 5	brtxp 31987	. 2 |- ( <. X , Y >. ( ( A o. ( 1st |` ( _V X. _V ) ) ) (x) ( B o. ( 2nd |` ( _V X. _V ) ) ) ) <. Z , W >. <-> ( <. X , Y >. ( A o. ( 1st |` ( _V X. _V ) ) ) Z /\ <. X , Y >. ( B o. ( 2nd |` ( _V X. _V ) ) ) W ) )
7	3, 4	brco 5292	. . . 4 |- ( <. X , Y >. ( A o. ( 1st |` ( _V X. _V ) ) ) Z <-> E. x ( <. X , Y >. ( 1st |` ( _V X. _V ) ) x /\ x A Z ) )
8		brpprod.1	. . . . . . . . 9 |- X e. _V
9		brpprod.2	. . . . . . . . 9 |- Y e. _V
10	8, 9	opelvv 5166	. . . . . . . 8 |- <. X , Y >. e. ( _V X. _V )
11		vex 3203	. . . . . . . . 9 |- x e. _V
12	11	brres 5402	. . . . . . . 8 |- ( <. X , Y >. ( 1st |` ( _V X. _V ) ) x <-> ( <. X , Y >. 1st x /\ <. X , Y >. e. ( _V X. _V ) ) )
13	10, 12	mpbiran2 954	. . . . . . 7 |- ( <. X , Y >. ( 1st |` ( _V X. _V ) ) x <-> <. X , Y >. 1st x )
14	8, 9	br1steq 31670	. . . . . . 7 |- ( <. X , Y >. 1st x <-> x = X )
15	13, 14	bitri 264	. . . . . 6 |- ( <. X , Y >. ( 1st |` ( _V X. _V ) ) x <-> x = X )
16	15	anbi1i 731	. . . . 5 |- ( ( <. X , Y >. ( 1st |` ( _V X. _V ) ) x /\ x A Z ) <-> ( x = X /\ x A Z ) )
17	16	exbii 1774	. . . 4 |- ( E. x ( <. X , Y >. ( 1st |` ( _V X. _V ) ) x /\ x A Z ) <-> E. x ( x = X /\ x A Z ) )
18		breq1 4656	. . . . 5 |- ( x = X -> ( x A Z <-> X A Z ) )
19	8, 18	ceqsexv 3242	. . . 4 |- ( E. x ( x = X /\ x A Z ) <-> X A Z )
20	7, 17, 19	3bitri 286	. . 3 |- ( <. X , Y >. ( A o. ( 1st |` ( _V X. _V ) ) ) Z <-> X A Z )
21	3, 5	brco 5292	. . . 4 |- ( <. X , Y >. ( B o. ( 2nd |` ( _V X. _V ) ) ) W <-> E. y ( <. X , Y >. ( 2nd |` ( _V X. _V ) ) y /\ y B W ) )
22		vex 3203	. . . . . . . . 9 |- y e. _V
23	22	brres 5402	. . . . . . . 8 |- ( <. X , Y >. ( 2nd |` ( _V X. _V ) ) y <-> ( <. X , Y >. 2nd y /\ <. X , Y >. e. ( _V X. _V ) ) )
24	10, 23	mpbiran2 954	. . . . . . 7 |- ( <. X , Y >. ( 2nd |` ( _V X. _V ) ) y <-> <. X , Y >. 2nd y )
25	8, 9	br2ndeq 31671	. . . . . . 7 |- ( <. X , Y >. 2nd y <-> y = Y )
26	24, 25	bitri 264	. . . . . 6 |- ( <. X , Y >. ( 2nd |` ( _V X. _V ) ) y <-> y = Y )
27	26	anbi1i 731	. . . . 5 |- ( ( <. X , Y >. ( 2nd |` ( _V X. _V ) ) y /\ y B W ) <-> ( y = Y /\ y B W ) )
28	27	exbii 1774	. . . 4 |- ( E. y ( <. X , Y >. ( 2nd |` ( _V X. _V ) ) y /\ y B W ) <-> E. y ( y = Y /\ y B W ) )
29		breq1 4656	. . . . 5 |- ( y = Y -> ( y B W <-> Y B W ) )
30	9, 29	ceqsexv 3242	. . . 4 |- ( E. y ( y = Y /\ y B W ) <-> Y B W )
31	21, 28, 30	3bitri 286	. . 3 |- ( <. X , Y >. ( B o. ( 2nd |` ( _V X. _V ) ) ) W <-> Y B W )
32	20, 31	anbi12i 733	. 2 |- ( ( <. X , Y >. ( A o. ( 1st |` ( _V X. _V ) ) ) Z /\ <. X , Y >. ( B o. ( 2nd |` ( _V X. _V ) ) ) W ) <-> ( X A Z /\ Y B W ) )
33	2, 6, 32	3bitri 286	1 |- ( <. X , Y >.pprod ( A , B ) <. Z , W >. <-> ( X A Z /\ Y B W ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.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