Theorem brpprod3a 31993

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:   z, w, R   w, S, z   w, X, z   w, Y, z   w, Z, z

Proof of Theorem brpprod3a

Step	Hyp	Ref	Expression
1		pprodss4v 31991	. . . . . . 7 |- pprod ( R , S ) C_ ( ( _V X. _V ) X. ( _V X. _V ) )
2	1	brel 5168	. . . . . 6 |- ( <. X , Y >.pprod ( R , S ) Z -> ( <. X , Y >. e. ( _V X. _V ) /\ Z e. ( _V X. _V ) ) )
3	2	simprd 479	. . . . 5 |- ( <. X , Y >.pprod ( R , S ) Z -> Z e. ( _V X. _V ) )
4		elvv 5177	. . . . 5 |- ( Z e. ( _V X. _V ) <-> E. z E. w Z = <. z , w >. )
5	3, 4	sylib 208	. . . 4 |- ( <. X , Y >.pprod ( R , S ) Z -> E. z E. w Z = <. z , w >. )
6	5	pm4.71ri 665	. . 3 |- ( <. X , Y >.pprod ( R , S ) Z <-> ( E. z E. w Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) )
7		19.41vv 1915	. . 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 267	. 2 |- ( <. X , Y >.pprod ( R , S ) Z <-> E. z E. w ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) )
9		breq2 4657	. . . 4 |- ( Z = <. z , w >. -> ( <. X , Y >.pprod ( R , S ) Z <-> <. X , Y >.pprod ( R , S ) <. z , w >. ) )
10	9	pm5.32i 669	. . 3 |- ( ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) Z ) <-> ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) )
11	10	2exbii 1775	. 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 3203	. . . . . 6 |- z e. _V
15		vex 3203	. . . . . 6 |- w e. _V
16	12, 13, 14, 15	brpprod 31992	. . . . 5 |- ( <. X , Y >.pprod ( R , S ) <. z , w >. <-> ( X R z /\ Y S w ) )
17	16	anbi2i 730	. . . 4 |- ( ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) <-> ( Z = <. z , w >. /\ ( X R z /\ Y S w ) ) )
18		3anass 1042	. . . 4 |- ( ( Z = <. z , w >. /\ X R z /\ Y S w ) <-> ( Z = <. z , w >. /\ ( X R z /\ Y S w ) ) )
19	17, 18	bitr4i 267	. . 3 |- ( ( Z = <. z , w >. /\ <. X , Y >.pprod ( R , S ) <. z , w >. ) <-> ( Z = <. z , w >. /\ X R z /\ Y S w ) )
20	19	2exbii 1775	. 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 286	1 |- ( <. X , Y >.pprod ( R , S ) Z <-> E. z E. w ( Z = <. z , w >. /\ X R z /\ Y S w ) )

Syntax hints:   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   X. cxp 5112  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:  brpprod3b  31994  brapply  32045  dfrdg4  32058