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Theorem brtxp 32293
Description: Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 32291, 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 32267 . . 3 (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
21breqi 4810 . 2 (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋(((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩)
3 brin 4856 . 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 5081 . . . . 5 𝑌, 𝑍⟩ ∈ V
64, 5brco 5448 . . . 4 (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ ∃𝑦(𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩))
7 ancom 465 . . . . . 6 ((𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐴𝑦))
8 vex 3343 . . . . . . . . 9 𝑦 ∈ V
98, 5brcnv 5460 . . . . . . . 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 5323 . . . . . . . . 9 𝑌, 𝑍⟩ ∈ (V × V)
138brres 5560 . . . . . . . . 9 (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ (⟨𝑌, 𝑍⟩1st 𝑦 ∧ ⟨𝑌, 𝑍⟩ ∈ (V × V)))
1412, 13mpbiran2 992 . . . . . . . 8 (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ ⟨𝑌, 𝑍⟩1st 𝑦)
1510, 11br1steq 31977 . . . . . . . 8 (⟨𝑌, 𝑍⟩1st 𝑦𝑦 = 𝑌)
169, 14, 153bitri 286 . . . . . . 7 (𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑦 = 𝑌)
1716anbi1i 733 . . . . . 6 ((𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐴𝑦) ↔ (𝑦 = 𝑌𝑋𝐴𝑦))
187, 17bitri 264 . . . . 5 ((𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦 = 𝑌𝑋𝐴𝑦))
1918exbii 1923 . . . 4 (∃𝑦(𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑦(𝑦 = 𝑌𝑋𝐴𝑦))
20 breq2 4808 . . . . 5 (𝑦 = 𝑌 → (𝑋𝐴𝑦𝑋𝐴𝑌))
2110, 20ceqsexv 3382 . . . 4 (∃𝑦(𝑦 = 𝑌𝑋𝐴𝑦) ↔ 𝑋𝐴𝑌)
226, 19, 213bitri 286 . . 3 (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐴𝑌)
234, 5brco 5448 . . . 4 (𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ ∃𝑧(𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩))
24 ancom 465 . . . . . 6 ((𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐵𝑧))
25 vex 3343 . . . . . . . . 9 𝑧 ∈ V
2625, 5brcnv 5460 . . . . . . . 8 (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧)
2725brres 5560 . . . . . . . . 9 (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ (⟨𝑌, 𝑍⟩2nd 𝑧 ∧ ⟨𝑌, 𝑍⟩ ∈ (V × V)))
2812, 27mpbiran2 992 . . . . . . . 8 (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ ⟨𝑌, 𝑍⟩2nd 𝑧)
2910, 11br2ndeq 31978 . . . . . . . 8 (⟨𝑌, 𝑍⟩2nd 𝑧𝑧 = 𝑍)
3026, 28, 293bitri 286 . . . . . . 7 (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑧 = 𝑍)
3130anbi1i 733 . . . . . 6 ((𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐵𝑧) ↔ (𝑧 = 𝑍𝑋𝐵𝑧))
3224, 31bitri 264 . . . . 5 ((𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧 = 𝑍𝑋𝐵𝑧))
3332exbii 1923 . . . 4 (∃𝑧(𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑧(𝑧 = 𝑍𝑋𝐵𝑧))
34 breq2 4808 . . . . 5 (𝑧 = 𝑍 → (𝑋𝐵𝑧𝑋𝐵𝑍))
3511, 34ceqsexv 3382 . . . 4 (∃𝑧(𝑧 = 𝑍𝑋𝐵𝑧) ↔ 𝑋𝐵𝑍)
3623, 33, 353bitri 286 . . 3 (𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐵𝑍)
3722, 36anbi12i 735 . 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 383   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  cin 3714  cop 4327   class class class wbr 4804   × cxp 5264  ccnv 5265  cres 5268  ccom 5270  1st c1st 7331  2nd c2nd 7332  ctxp 32243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-1st 7333  df-2nd 7334  df-txp 32267
This theorem is referenced by:  brtxp2  32294  pprodss4v  32297  brpprod  32298  brsset  32302  brtxpsd  32307  elfuns  32328
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