Theorem brpprod3a 30702

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 30700	. . . . . . 7 |- pprod ( R , S ) C_ ( ( _V X. _V ) X. ( _V X. _V ) )
2	1	brel 4902	. . . . . 6 |- ( <. X , Y >.pprod ( R , S ) Z -> ( <. X , Y >. e. ( _V X. _V ) /\ Z e. ( _V X. _V ) ) )
3	2	simprd 469	. . . . 5 |- ( <. X , Y >.pprod ( R , S ) Z -> Z e. ( _V X. _V ) )
4		elvv 4912	. . . . 5 |- ( Z e. ( _V X. _V ) <-> E. z E. w Z = <. z , w >. )
5	3, 4	sylib 201	. . . 4 |- ( <. X , Y >.pprod ( R , S ) Z -> E. z E. w Z = <. z , w >. )
6	5	pm4.71ri 643	. . 3 |- ( <. X , Y >.pprod ( R , S ) Z <-> ( E. z E. w Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) )
7		19.41vv 1842	. . 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 260	. 2 |- ( <. X , Y >.pprod ( R , S ) Z <-> E. z E. w ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) )
9		breq2 4420	. . . 4 |- ( Z = <. z , w >. -> ( <. X , Y >.pprod ( R , S ) Z <-> <. X , Y >.pprod ( R , S ) <. z , w >. ) )
10	9	pm5.32i 647	. . 3 |- ( ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) <-> ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) )
11	10	2exbii 1730	. 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 3060	. . . . . 6 |- z e. _V
15		vex 3060	. . . . . 6 |- w e. _V
16	12, 13, 14, 15	brpprod 30701	. . . . 5 |- ( <. X , Y >.pprod ( R , S ) <. z , w >. <-> ( X R z /\ Y S w ) )
17	16	anbi2i 705	. . . 4 |- ( ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) <-> ( Z = <. z , w >. /\ ( X R z /\ Y S w ) ) )
18		3anass 995	. . . 4 |- ( ( Z = <. z , w >. /\ X R z /\ Y S w ) <-> ( Z = <. z , w >. /\ ( X R z /\ Y S w ) ) )
19	17, 18	bitr4i 260	. . 3 |- ( ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) <-> ( Z = <. z , w >. /\ X R z /\ Y S w ) )
20	19	2exbii 1730	. 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 279	1 |- ( <. X , Y >.pprod ( R , S ) Z <-> E. z E. w ( Z = <. z , w >. /\ X R z /\ Y S w ) )

Syntax hints:   <-> wb 189   /\ wa 375   /\ w3a 991   = wceq 1455   E.wex 1674   e. wcel 1898   _Vcvv 3057   <.cop 3986   class class class wbr 4416   X. cxp 4851  pprodcpprod 30646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fo 5607  df-fv 5609  df-1st 6820  df-2nd 6821  df-txp 30669  df-pprod 30670

This theorem is referenced by:  brpprod3b  30703  brapply  30754  dfrdg4  30767