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Theorem brtxp 31626
Description: Characterize a trinary relationship over a tail Cartesian product. Together with txpss3v 31624, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.)
Hypotheses
Ref Expression
brtxp.1 𝑋 ∈ V
brtxp.2 𝑌 ∈ V
brtxp.3 𝑍 ∈ V
Assertion
Ref Expression
brtxp (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌𝑋𝐵𝑍))

Proof of Theorem brtxp
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-txp 31599 . . 3 (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
21breqi 4619 . 2 (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋(((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩)
3 brin 4664 . 2 (𝑋(((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩ ↔ (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ∧ 𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩))
4 brtxp.1 . . . . 5 𝑋 ∈ V
5 opex 4893 . . . . 5 𝑌, 𝑍⟩ ∈ V
64, 5brco 5252 . . . 4 (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ ∃𝑦(𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩))
7 ancom 466 . . . . . 6 ((𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐴𝑦))
8 vex 3189 . . . . . . . . 9 𝑦 ∈ V
98, 5brcnv 5265 . . . . . . . 8 (𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦)
10 brtxp.2 . . . . . . . . . 10 𝑌 ∈ V
11 brtxp.3 . . . . . . . . . 10 𝑍 ∈ V
1210, 11opelvv 5126 . . . . . . . . 9 𝑌, 𝑍⟩ ∈ (V × V)
138brres 5362 . . . . . . . . 9 (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ (⟨𝑌, 𝑍⟩1st 𝑦 ∧ ⟨𝑌, 𝑍⟩ ∈ (V × V)))
1412, 13mpbiran2 953 . . . . . . . 8 (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ ⟨𝑌, 𝑍⟩1st 𝑦)
1510, 11, 8br1steq 31371 . . . . . . . 8 (⟨𝑌, 𝑍⟩1st 𝑦𝑦 = 𝑌)
169, 14, 153bitri 286 . . . . . . 7 (𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑦 = 𝑌)
1716anbi1i 730 . . . . . 6 ((𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐴𝑦) ↔ (𝑦 = 𝑌𝑋𝐴𝑦))
187, 17bitri 264 . . . . 5 ((𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦 = 𝑌𝑋𝐴𝑦))
1918exbii 1771 . . . 4 (∃𝑦(𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑦(𝑦 = 𝑌𝑋𝐴𝑦))
20 breq2 4617 . . . . 5 (𝑦 = 𝑌 → (𝑋𝐴𝑦𝑋𝐴𝑌))
2110, 20ceqsexv 3228 . . . 4 (∃𝑦(𝑦 = 𝑌𝑋𝐴𝑦) ↔ 𝑋𝐴𝑌)
226, 19, 213bitri 286 . . 3 (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐴𝑌)
234, 5brco 5252 . . . 4 (𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ ∃𝑧(𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩))
24 ancom 466 . . . . . 6 ((𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐵𝑧))
25 vex 3189 . . . . . . . . 9 𝑧 ∈ V
2625, 5brcnv 5265 . . . . . . . 8 (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧)
2725brres 5362 . . . . . . . . 9 (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ (⟨𝑌, 𝑍⟩2nd 𝑧 ∧ ⟨𝑌, 𝑍⟩ ∈ (V × V)))
2812, 27mpbiran2 953 . . . . . . . 8 (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ ⟨𝑌, 𝑍⟩2nd 𝑧)
2910, 11, 25br2ndeq 31372 . . . . . . . 8 (⟨𝑌, 𝑍⟩2nd 𝑧𝑧 = 𝑍)
3026, 28, 293bitri 286 . . . . . . 7 (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑧 = 𝑍)
3130anbi1i 730 . . . . . 6 ((𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐵𝑧) ↔ (𝑧 = 𝑍𝑋𝐵𝑧))
3224, 31bitri 264 . . . . 5 ((𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧 = 𝑍𝑋𝐵𝑧))
3332exbii 1771 . . . 4 (∃𝑧(𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑧(𝑧 = 𝑍𝑋𝐵𝑧))
34 breq2 4617 . . . . 5 (𝑧 = 𝑍 → (𝑋𝐵𝑧𝑋𝐵𝑍))
3511, 34ceqsexv 3228 . . . 4 (∃𝑧(𝑧 = 𝑍𝑋𝐵𝑧) ↔ 𝑋𝐵𝑍)
3623, 33, 353bitri 286 . . 3 (𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐵𝑍)
3722, 36anbi12i 732 . 2 ((𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ∧ 𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩) ↔ (𝑋𝐴𝑌𝑋𝐵𝑍))
382, 3, 373bitri 286 1 (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌𝑋𝐵𝑍))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186  cin 3554  cop 4154   class class class wbr 4613   × cxp 5072  ccnv 5073  cres 5076  ccom 5078  1st c1st 7111  2nd c2nd 7112  ctxp 31575
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-1st 7113  df-2nd 7114  df-txp 31599
This theorem is referenced by:  brtxp2  31627  pprodss4v  31630  brpprod  31631  brsset  31635  brtxpsd  31640  elfuns  31661
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