Theorem brpprod3a 29689

Description: Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brpprod3.1	|- X e. _V
brpprod3.2	|- Y e. _V
brpprod3.3	|- Z e. _V

Assertion

Ref	Expression
brpprod3a	|- ( <. X , Y >.pprod ( R , S ) Z <-> E. z E. w ( Z = <. z , w >. /\ X R z /\ Y S w ) )

Distinct variable groups:   w, R, z   w, S, z   w, X, z   w, Y, z   w, Z, z

Proof of Theorem brpprod3a

Step	Hyp	Ref	Expression
1		pprodss4v 29687	. . . . . . 7 |- pprod ( R , S ) C_ ( ( _V X. _V ) X. ( _V X. _V ) )
2	1	brel 4962	. . . . . 6 |- ( <. X , Y >.pprod ( R , S ) Z -> ( <. X , Y >. e. ( _V X. _V ) /\ Z e. ( _V X. _V ) ) )
3	2	simprd 461	. . . . 5 |- ( <. X , Y >.pprod ( R , S ) Z -> Z e. ( _V X. _V ) )
4		elvv 4972	. . . . 5 |- ( Z e. ( _V X. _V ) <-> E. z E. w Z = <. z , w >. )
5	3, 4	sylib 196	. . . 4 |- ( <. X , Y >.pprod ( R , S ) Z -> E. z E. w Z = <. z , w >. )
6	5	pm4.71ri 631	. . 3 |- ( <. X , Y >.pprod ( R , S ) Z <-> ( E. z E. w Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) )
7		19.41vv 1780	. . 3 |- ( E. z E. w ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) <-> ( E. z E. w Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) )
8	6, 7	bitr4i 252	. 2 |- ( <. X , Y >.pprod ( R , S ) Z <-> E. z E. w ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) )
9		breq2 4371	. . . 4 |- ( Z = <. z , w >. -> ( <. X , Y >.pprod ( R , S ) Z <-> <. X , Y >.pprod ( R , S ) <. z , w >. ) )
10	9	pm5.32i 635	. . 3 |- ( ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) <-> ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) )
11	10	2exbii 1676	. 2 |- ( E. z E. w ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) <-> E. z E. w ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) )
12		brpprod3.1	. . . . . 6 |- X e. _V
13		brpprod3.2	. . . . . 6 |- Y e. _V
14		vex 3037	. . . . . 6 |- z e. _V
15		vex 3037	. . . . . 6 |- w e. _V
16	12, 13, 14, 15	brpprod 29688	. . . . 5 |- ( <. X , Y >.pprod ( R , S ) <. z , w >. <-> ( X R z /\ Y S w ) )
17	16	anbi2i 692	. . . 4 |- ( ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) <-> ( Z = <. z , w >. /\ ( X R z /\ Y S w ) ) )
18		3anass 975	. . . 4 |- ( ( Z = <. z , w >. /\ X R z /\ Y S w ) <-> ( Z = <. z , w >. /\ ( X R z /\ Y S w ) ) )
19	17, 18	bitr4i 252	. . 3 |- ( ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) <-> ( Z = <. z , w >. /\ X R z /\ Y S w ) )
20	19	2exbii 1676	. 2 |- ( E. z E. w ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) <-> E. z E. w ( Z = <. z , w >. /\ X R z /\ Y S w ) )
21	8, 11, 20	3bitri 271	1 |- ( <. X , Y >.pprod ( R , S ) Z <-> E. z E. w ( Z = <. z , w >. /\ X R z /\ Y S w ) )

Syntax hints:   <-> wb 184   /\ wa 367   /\ w3a 971   = wceq 1399   E.wex 1620   e. wcel 1826   _Vcvv 3034   <.cop 3950   class class class wbr 4367   X. cxp 4911  pprodcpprod 29633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fo 5502  df-fv 5504  df-1st 6699  df-2nd 6700  df-txp 29656  df-pprod 29657

This theorem is referenced by:  brpprod3b  29690  brapply  29741  dfrdg4  29753