Theorem brtxp 30473

Description: Characterize a trinary relationship over a tail Cartesian product. Together with txpss3v 30471, 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 30446	. . 3 |- ( A (x) B ) = ( ( `' ( 1st |` ( _V X. _V ) ) o. A ) i^i ( `' ( 2nd |` ( _V X. _V ) ) o. B ) )
2	1	breqi 4423	. 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 4466	. 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 4677	. . . . 5 |- <. Y , Z >. e. _V
6	4, 5	brco 5016	. . . 4 |- ( X ( `' ( 1st |` ( _V X. _V ) ) o. A ) <. Y , Z >. <-> E. y ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) )
7		ancom 451	. . . . . 6 |- ( ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. /\ X A y ) )
8		vex 3081	. . . . . . . . 9 |- y e. _V
9	8, 5	brcnv 5028	. . . . . . . 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 4892	. . . . . . . . 9 |- <. Y , Z >. e. ( _V X. _V )
13	8	brres 5122	. . . . . . . . 9 |- ( <. Y , Z >. ( 1st |` ( _V X. _V ) ) y <-> ( <. Y , Z >. 1st y /\ <. Y , Z >. e. ( _V X. _V ) ) )
14	12, 13	mpbiran2 927	. . . . . . . 8 |- ( <. Y , Z >. ( 1st |` ( _V X. _V ) ) y <-> <. Y , Z >. 1st y )
15	10, 11, 8	br1steq 30242	. . . . . . . 8 |- ( <. Y , Z >. 1st y <-> y = Y )
16	9, 14, 15	3bitri 274	. . . . . . 7 |- ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. <-> y = Y )
17	16	anbi1i 699	. . . . . 6 |- ( ( y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. /\ X A y ) <-> ( y = Y /\ X A y ) )
18	7, 17	bitri 252	. . . . 5 |- ( ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( y = Y /\ X A y ) )
19	18	exbii 1712	. . . 4 |- ( E. y ( X A y /\ y `' ( 1st |` ( _V X. _V ) ) <. Y , Z >. ) <-> E. y ( y = Y /\ X A y ) )
20		breq2 4421	. . . . 5 |- ( y = Y -> ( X A y <-> X A Y ) )
21	10, 20	ceqsexv 3115	. . . 4 |- ( E. y ( y = Y /\ X A y ) <-> X A Y )
22	6, 19, 21	3bitri 274	. . 3 |- ( X ( `' ( 1st |` ( _V X. _V ) ) o. A ) <. Y , Z >. <-> X A Y )
23	4, 5	brco 5016	. . . 4 |- ( X ( `' ( 2nd |` ( _V X. _V ) ) o. B ) <. Y , Z >. <-> E. z ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) )
24		ancom 451	. . . . . 6 |- ( ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. /\ X B z ) )
25		vex 3081	. . . . . . . . 9 |- z e. _V
26	25, 5	brcnv 5028	. . . . . . . 8 |- ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. <-> <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z )
27	25	brres 5122	. . . . . . . . 9 |- ( <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z <-> ( <. Y , Z >. 2nd z /\ <. Y , Z >. e. ( _V X. _V ) ) )
28	12, 27	mpbiran2 927	. . . . . . . 8 |- ( <. Y , Z >. ( 2nd |` ( _V X. _V ) ) z <-> <. Y , Z >. 2nd z )
29	10, 11, 25	br2ndeq 30243	. . . . . . . 8 |- ( <. Y , Z >. 2nd z <-> z = Z )
30	26, 28, 29	3bitri 274	. . . . . . 7 |- ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. <-> z = Z )
31	30	anbi1i 699	. . . . . 6 |- ( ( z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. /\ X B z ) <-> ( z = Z /\ X B z ) )
32	24, 31	bitri 252	. . . . 5 |- ( ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> ( z = Z /\ X B z ) )
33	32	exbii 1712	. . . 4 |- ( E. z ( X B z /\ z `' ( 2nd |` ( _V X. _V ) ) <. Y , Z >. ) <-> E. z ( z = Z /\ X B z ) )
34		breq2 4421	. . . . 5 |- ( z = Z -> ( X B z <-> X B Z ) )
35	11, 34	ceqsexv 3115	. . . 4 |- ( E. z ( z = Z /\ X B z ) <-> X B Z )
36	23, 33, 35	3bitri 274	. . 3 |- ( X ( `' ( 2nd |` ( _V X. _V ) ) o. B ) <. Y , Z >. <-> X B Z )
37	22, 36	anbi12i 701	. 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 274	1 |- ( X ( A (x) B ) <. Y , Z >. <-> ( X A Y /\ X B Z ) )

Syntax hints:   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1659   e. wcel 1867   _Vcvv 3078   i^i cin 3432   <.cop 3999   class class class wbr 4417   X. cxp 4843   `'ccnv 4844   |` cres 4847   o. ccom 4849   1stc1st 6796   2ndc2nd 6797   (x) ctxp 30422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-fo 5598  df-fv 5600  df-1st 6798  df-2nd 6799  df-txp 30446

This theorem is referenced by:  brtxp2  30474  pprodss4v  30477  brpprod  30478  brsset  30482  brtxpsd  30487  elfuns  30508