Theorem brtxp 31987

Description: Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 31985, 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 31961	. . 3 |- ( A (x) B ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) )
2	1	breqi 4659	. 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 4704	. 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 4932	. . . . 5 |- <. Y , Z >. e. _V
6	4, 5	brco 5292	. . . 4 |- ( X ( `' ( 1st |` ( _V X. _V ) ) o. A ) <. Y , Z >. <-> E. y ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) )
7		ancom 466	. . . . . 6 |- ( ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. /\ X A y ) )
8		vex 3203	. . . . . . . . 9 |- y e. _V
9	8, 5	brcnv 5305	. . . . . . . 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 5166	. . . . . . . . 9 |- <. Y , Z >. e. ( _V X. _V )
13	8	brres 5402	. . . . . . . . 9 |- ( <. Y , Z >. ( 1st |` ( _V X. _V ) ) y <-> ( <. Y , Z >. 1st y /\ <. Y , Z >. e. ( _V X. _V ) ) )
14	12, 13	mpbiran2 954	. . . . . . . 8 |- ( <. Y , Z >. ( 1st |` ( _V X. _V ) ) y <-> <. Y , Z >. 1st y )
15	10, 11	br1steq 31670	. . . . . . . 8 |- ( <. Y , Z >. 1st y <-> y = Y )
16	9, 14, 15	3bitri 286	. . . . . . 7 |- ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. <-> y = Y )
17	16	anbi1i 731	. . . . . 6 |- ( ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. /\ X A y ) <-> ( y = Y /\ X A y ) )
18	7, 17	bitri 264	. . . . 5 |- ( ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( y = Y /\ X A y ) )
19	18	exbii 1774	. . . 4 |- ( E. y ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> E. y ( y = Y /\ X A y ) )
20		breq2 4657	. . . . 5 |- ( y = Y -> ( X A y <-> X A Y ) )
21	10, 20	ceqsexv 3242	. . . 4 |- ( E. y ( y = Y /\ X A y ) <-> X A Y )
22	6, 19, 21	3bitri 286	. . 3 |- ( X ( `' ( 1st |` ( _V X. _V ) ) o. A ) <. Y , Z >. <-> X A Y )
23	4, 5	brco 5292	. . . 4 |- ( X ( `' ( 2nd |` ( _V X. _V ) ) o. B ) <. Y , Z >. <-> E. z ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) )
24		ancom 466	. . . . . 6 |- ( ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. /\ X B z ) )
25		vex 3203	. . . . . . . . 9 |- z e. _V
26	25, 5	brcnv 5305	. . . . . . . 8 |- ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. <-> <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z )
27	25	brres 5402	. . . . . . . . 9 |- ( <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z <-> ( <. Y , Z >. 2nd z /\ <. Y , Z >. e. ( _V X. _V ) ) )
28	12, 27	mpbiran2 954	. . . . . . . 8 |- ( <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z <-> <. Y , Z >. 2nd z )
29	10, 11	br2ndeq 31671	. . . . . . . 8 |- ( <. Y , Z >. 2nd z <-> z = Z )
30	26, 28, 29	3bitri 286	. . . . . . 7 |- ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. <-> z = Z )
31	30	anbi1i 731	. . . . . 6 |- ( ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. /\ X B z ) <-> ( z = Z /\ X B z ) )
32	24, 31	bitri 264	. . . . 5 |- ( ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( z = Z /\ X B z ) )
33	32	exbii 1774	. . . 4 |- ( E. z ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> E. z ( z = Z /\ X B z ) )
34		breq2 4657	. . . . 5 |- ( z = Z -> ( X B z <-> X B Z ) )
35	11, 34	ceqsexv 3242	. . . 4 |- ( E. z ( z = Z /\ X B z ) <-> X B Z )
36	23, 33, 35	3bitri 286	. . 3 |- ( X ( `' ( 2nd |` ( _V X. _V ) ) o. B ) <. Y , Z >. <-> X B Z )
37	22, 36	anbi12i 733	. 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 286	1 |- ( X ( A (x) B ) <. Y , Z >. <-> ( X A Y /\ X B Z ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   i^i cin 3573   <.cop 4183   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   |` cres 5116   o. ccom 5118   1stc1st 7166   2ndc2nd 7167   (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:  brtxp2  31988  pprodss4v  31991  brpprod  31992  brsset  31996  brtxpsd  32001  elfuns  32022