Theorem brtxp 29383

Description: Characterize a trinary relationship over a tail Cartesian product. Together with txpss3v 29381, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)

Hypotheses

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

Assertion

Ref	Expression
brtxp	|- ( X ( A (x) B ) <. Y , Z >. <-> ( X A Y /\ X B Z ) )

Proof of Theorem brtxp

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

Step	Hyp	Ref	Expression
1		df-txp 29356	. . 3 |- ( A (x) B ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) )
2	1	breqi 4453	. 2 |- ( X ( A (x) B ) <. Y , Z >. <-> X ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) ) <. Y , Z >. )
3		brin 4496	. 2 |- ( X ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) ) <. Y , Z >. <-> ( X ( `' ( 1st |` ( _V X. _V ) ) o. A ) <. Y , Z >. /\ X ( `' ( 2nd |` ( _V X. _V ) ) o. B ) <. Y , Z >. ) )
4		brtxp.1	. . . . 5 |- X e. _V
5		opex 4711	. . . . 5 |- <. Y , Z >. e. _V
6	4, 5	brco 5173	. . . 4 |- ( X ( `' ( 1st |` ( _V X. _V ) ) o. A ) <. Y , Z >. <-> E. y ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) )
7		ancom 450	. . . . . 6 |- ( ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. /\ X A y ) )
8		vex 3116	. . . . . . . . 9 |- y e. _V
9	8, 5	brcnv 5185	. . . . . . . 8 |- ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. <-> <. Y , Z >. ( 1st |` ( _V X. _V ) ) y )
10		brtxp.2	. . . . . . . . . 10 |- Y e. _V
11		brtxp.3	. . . . . . . . . 10 |- Z e. _V
12	10, 11	opelvv 5046	. . . . . . . . 9 |- <. Y , Z >. e. ( _V X. _V )
13	8	brres 5280	. . . . . . . . 9 |- ( <. Y , Z >. ( 1st |` ( _V X. _V ) ) y <-> ( <. Y , Z >. 1st y /\ <. Y , Z >. e. ( _V X. _V ) ) )
14	12, 13	mpbiran2 917	. . . . . . . 8 |- ( <. Y , Z >. ( 1st |` ( _V X. _V ) ) y <-> <. Y , Z >. 1st y )
15	10, 11, 8	br1steq 29057	. . . . . . . 8 |- ( <. Y , Z >. 1st y <-> y = Y )
16	9, 14, 15	3bitri 271	. . . . . . 7 |- ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. <-> y = Y )
17	16	anbi1i 695	. . . . . 6 |- ( ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. /\ X A y ) <-> ( y = Y /\ X A y ) )
18	7, 17	bitri 249	. . . . 5 |- ( ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( y = Y /\ X A y ) )
19	18	exbii 1644	. . . 4 |- ( E. y ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> E. y ( y = Y /\ X A y ) )
20		breq2 4451	. . . . 5 |- ( y = Y -> ( X A y <-> X A Y ) )
21	10, 20	ceqsexv 3150	. . . 4 |- ( E. y ( y = Y /\ X A y ) <-> X A Y )
22	6, 19, 21	3bitri 271	. . 3 |- ( X ( `' ( 1st |` ( _V X. _V ) ) o. A ) <. Y , Z >. <-> X A Y )
23	4, 5	brco 5173	. . . 4 |- ( X ( `' ( 2nd |` ( _V X. _V ) ) o. B ) <. Y , Z >. <-> E. z ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) )
24		ancom 450	. . . . . 6 |- ( ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. /\ X B z ) )
25		vex 3116	. . . . . . . . 9 |- z e. _V
26	25, 5	brcnv 5185	. . . . . . . 8 |- ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. <-> <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z )
27	25	brres 5280	. . . . . . . . 9 |- ( <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z <-> ( <. Y , Z >. 2nd z /\ <. Y , Z >. e. ( _V X. _V ) ) )
28	12, 27	mpbiran2 917	. . . . . . . 8 |- ( <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z <-> <. Y , Z >. 2nd z )
29	10, 11, 25	br2ndeq 29058	. . . . . . . 8 |- ( <. Y , Z >. 2nd z <-> z = Z )
30	26, 28, 29	3bitri 271	. . . . . . 7 |- ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. <-> z = Z )
31	30	anbi1i 695	. . . . . 6 |- ( ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. /\ X B z ) <-> ( z = Z /\ X B z ) )
32	24, 31	bitri 249	. . . . 5 |- ( ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( z = Z /\ X B z ) )
33	32	exbii 1644	. . . 4 |- ( E. z ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> E. z ( z = Z /\ X B z ) )
34		breq2 4451	. . . . 5 |- ( z = Z -> ( X B z <-> X B Z ) )
35	11, 34	ceqsexv 3150	. . . 4 |- ( E. z ( z = Z /\ X B z ) <-> X B Z )
36	23, 33, 35	3bitri 271	. . 3 |- ( X ( `' ( 2nd |` ( _V X. _V ) ) o. B ) <. Y , Z >. <-> X B Z )
37	22, 36	anbi12i 697	. 2 |- ( ( X ( `' ( 1st |` ( _V X. _V ) ) o. A ) <. Y , Z >. /\ X ( `' ( 2nd |` ( _V X. _V ) ) o. B ) <. Y , Z >. ) <-> ( X A Y /\ X B Z ) )
38	2, 3, 37	3bitri 271	1 |- ( X ( A (x) B ) <. Y , Z >. <-> ( X A Y /\ X B Z ) )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   _Vcvv 3113   i^i cin 3475   <.cop 4033   class class class wbr 4447   X. cxp 4997   `'ccnv 4998   |` cres 5001   o. ccom 5003   1stc1st 6783   2ndc2nd 6784   (x) ctxp 29332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fo 5594  df-fv 5596  df-1st 6785  df-2nd 6786  df-txp 29356

This theorem is referenced by:  brtxp2  29384  pprodss4v  29387  brpprod  29388  brsset  29392  brtxpsd  29397  elfuns  29418