Theorem brtxp2 31988

Description: The binary relation over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)

Hypothesis

Ref	Expression
brtxp2.1	|- A e. _V

Assertion

Ref	Expression
brtxp2	|- ( A ( R (x) S ) B <-> E. x E. y ( B = <. x , y >. /\ A R x /\ A S y ) )

Distinct variable groups:   x, A, y   x, B, y   x, R, y   x, S, y

Proof of Theorem brtxp2

Step	Hyp	Ref	Expression
1		txpss3v 31985	. . . . . . 7 |- ( R (x) S ) C_ ( _V X. ( _V X. _V ) )
2	1	brel 5168	. . . . . 6 |- ( A ( R (x) S ) B -> ( A e. _V /\ B e. ( _V X. _V ) ) )
3	2	simprd 479	. . . . 5 |- ( A ( R (x) S ) B -> B e. ( _V X. _V ) )
4		elvv 5177	. . . . 5 |- ( B e. ( _V X. _V ) <-> E. x E. y B = <. x , y >. )
5	3, 4	sylib 208	. . . 4 |- ( A ( R (x) S ) B -> E. x E. y B = <. x , y >. )
6	5	pm4.71ri 665	. . 3 |- ( A ( R (x) S ) B <-> ( E. x E. y B = <. x , y >. /\ A ( R (x) S ) B ) )
7		19.41vv 1915	. . 3 |- ( E. x E. y ( B = <. x , y >. /\ A ( R (x) S ) B ) <-> ( E. x E. y B = <. x , y >. /\ A ( R (x) S ) B ) )
8	6, 7	bitr4i 267	. 2 |- ( A ( R (x) S ) B <-> E. x E. y ( B = <. x , y >. /\ A ( R (x) S ) B ) )
9		breq2 4657	. . . 4 |- ( B = <. x , y >. -> ( A ( R (x) S ) B <-> A ( R (x) S ) <. x , y >. ) )
10	9	pm5.32i 669	. . 3 |- ( ( B = <. x , y >. /\ A ( R (x) S ) B ) <-> ( B = <. x , y >. /\ A ( R (x) S ) <. x , y >. ) )
11	10	2exbii 1775	. 2 |- ( E. x E. y ( B = <. x , y >. /\ A ( R (x) S ) B ) <-> E. x E. y ( B = <. x , y >. /\ A ( R (x) S ) <. x , y >. ) )
12		brtxp2.1	. . . . . 6 |- A e. _V
13		vex 3203	. . . . . 6 |- x e. _V
14		vex 3203	. . . . . 6 |- y e. _V
15	12, 13, 14	brtxp 31987	. . . . 5 |- ( A ( R (x) S ) <. x , y >. <-> ( A R x /\ A S y ) )
16	15	anbi2i 730	. . . 4 |- ( ( B = <. x , y >. /\ A ( R (x) S ) <. x , y >. ) <-> ( B = <. x , y >. /\ ( A R x /\ A S y ) ) )
17		3anass 1042	. . . 4 |- ( ( B = <. x , y >. /\ A R x /\ A S y ) <-> ( B = <. x , y >. /\ ( A R x /\ A S y ) ) )
18	16, 17	bitr4i 267	. . 3 |- ( ( B = <. x , y >. /\ A ( R (x) S ) <. x , y >. ) <-> ( B = <. x , y >. /\ A R x /\ A S y ) )
19	18	2exbii 1775	. 2 |- ( E. x E. y ( B = <. x , y >. /\ A ( R (x) S ) <. x , y >. ) <-> E. x E. y ( B = <. x , y >. /\ A R x /\ A S y ) )
20	8, 11, 19	3bitri 286	1 |- ( A ( R (x) S ) B <-> E. x E. y ( B = <. x , y >. /\ A R x /\ A S y ) )

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   (x) ctxp 31937

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

This theorem is referenced by:  brsuccf  32048  brrestrict  32056