Theorem brtxp 30640

Description: Characterize a trinary relationship over a tail Cartesian product. Together with txpss3v 30638, 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 30613	. . 3 |- ( A (x) B ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) )
2	1	breqi 4407	. 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 4451	. 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 4663	. . . . 5 |- <. Y , Z >. e. _V
6	4, 5	brco 5004	. . . 4 |- ( X ( `' ( 1st |` ( _V X. _V ) ) o. A ) <. Y , Z >. <-> E. y ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) )
7		ancom 452	. . . . . 6 |- ( ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. /\ X A y ) )
8		vex 3047	. . . . . . . . 9 |- y e. _V
9	8, 5	brcnv 5016	. . . . . . . 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 4880	. . . . . . . . 9 |- <. Y , Z >. e. ( _V X. _V )
13	8	brres 5110	. . . . . . . . 9 |- ( <. Y , Z >. ( 1st |` ( _V X. _V ) ) y <-> ( <. Y , Z >. 1st y /\ <. Y , Z >. e. ( _V X. _V ) ) )
14	12, 13	mpbiran2 929	. . . . . . . 8 |- ( <. Y , Z >. ( 1st |` ( _V X. _V ) ) y <-> <. Y , Z >. 1st y )
15	10, 11, 8	br1steq 30407	. . . . . . . 8 |- ( <. Y , Z >. 1st y <-> y = Y )
16	9, 14, 15	3bitri 275	. . . . . . 7 |- ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. <-> y = Y )
17	16	anbi1i 700	. . . . . 6 |- ( ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. /\ X A y ) <-> ( y = Y /\ X A y ) )
18	7, 17	bitri 253	. . . . 5 |- ( ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( y = Y /\ X A y ) )
19	18	exbii 1717	. . . 4 |- ( E. y ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> E. y ( y = Y /\ X A y ) )
20		breq2 4405	. . . . 5 |- ( y = Y -> ( X A y <-> X A Y ) )
21	10, 20	ceqsexv 3083	. . . 4 |- ( E. y ( y = Y /\ X A y ) <-> X A Y )
22	6, 19, 21	3bitri 275	. . 3 |- ( X ( `' ( 1st |` ( _V X. _V ) ) o. A ) <. Y , Z >. <-> X A Y )
23	4, 5	brco 5004	. . . 4 |- ( X ( `' ( 2nd |` ( _V X. _V ) ) o. B ) <. Y , Z >. <-> E. z ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) )
24		ancom 452	. . . . . 6 |- ( ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. /\ X B z ) )
25		vex 3047	. . . . . . . . 9 |- z e. _V
26	25, 5	brcnv 5016	. . . . . . . 8 |- ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. <-> <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z )
27	25	brres 5110	. . . . . . . . 9 |- ( <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z <-> ( <. Y , Z >. 2nd z /\ <. Y , Z >. e. ( _V X. _V ) ) )
28	12, 27	mpbiran2 929	. . . . . . . 8 |- ( <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z <-> <. Y , Z >. 2nd z )
29	10, 11, 25	br2ndeq 30408	. . . . . . . 8 |- ( <. Y , Z >. 2nd z <-> z = Z )
30	26, 28, 29	3bitri 275	. . . . . . 7 |- ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. <-> z = Z )
31	30	anbi1i 700	. . . . . 6 |- ( ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. /\ X B z ) <-> ( z = Z /\ X B z ) )
32	24, 31	bitri 253	. . . . 5 |- ( ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( z = Z /\ X B z ) )
33	32	exbii 1717	. . . 4 |- ( E. z ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> E. z ( z = Z /\ X B z ) )
34		breq2 4405	. . . . 5 |- ( z = Z -> ( X B z <-> X B Z ) )
35	11, 34	ceqsexv 3083	. . . 4 |- ( E. z ( z = Z /\ X B z ) <-> X B Z )
36	23, 33, 35	3bitri 275	. . 3 |- ( X ( `' ( 2nd |` ( _V X. _V ) ) o. B ) <. Y , Z >. <-> X B Z )
37	22, 36	anbi12i 702	. 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 275	1 |- ( X ( A (x) B ) <. Y , Z >. <-> ( X A Y /\ X B Z ) )

Syntax hints:   <-> wb 188   /\ wa 371   = wceq 1443   E.wex 1662   e. wcel 1886   _Vcvv 3044   i^i cin 3402   <.cop 3973   class class class wbr 4401   X. cxp 4831   `'ccnv 4832   |` cres 4835   o. ccom 4837   1stc1st 6788   2ndc2nd 6789   (x) ctxp 30589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589  df-1st 6790  df-2nd 6791  df-txp 30613

This theorem is referenced by:  brtxp2  30641  pprodss4v  30644  brpprod  30645  brsset  30649  brtxpsd  30654  elfuns  30675