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Theorem brtxp 32324
Description: Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 32322, 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 32298 . . 3 (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
21breqi 4793 . 2 (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋(((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩)
3 brin 4839 . 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 5061 . . . . 5 𝑌, 𝑍⟩ ∈ V
64, 5brco 5430 . . . 4 (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ ∃𝑦(𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩))
7 ancom 448 . . . . . 6 ((𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐴𝑦))
8 vex 3354 . . . . . . . . 9 𝑦 ∈ V
98, 5brcnv 5442 . . . . . . . 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 5305 . . . . . . . . 9 𝑌, 𝑍⟩ ∈ (V × V)
138brres 5542 . . . . . . . . 9 (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ (⟨𝑌, 𝑍⟩1st 𝑦 ∧ ⟨𝑌, 𝑍⟩ ∈ (V × V)))
1412, 13mpbiran2 689 . . . . . . . 8 (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ ⟨𝑌, 𝑍⟩1st 𝑦)
1510, 11br1steq 32008 . . . . . . . 8 (⟨𝑌, 𝑍⟩1st 𝑦𝑦 = 𝑌)
169, 14, 153bitri 286 . . . . . . 7 (𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑦 = 𝑌)
1716anbi1i 610 . . . . . 6 ((𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐴𝑦) ↔ (𝑦 = 𝑌𝑋𝐴𝑦))
187, 17bitri 264 . . . . 5 ((𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦 = 𝑌𝑋𝐴𝑦))
1918exbii 1924 . . . 4 (∃𝑦(𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑦(𝑦 = 𝑌𝑋𝐴𝑦))
20 breq2 4791 . . . . 5 (𝑦 = 𝑌 → (𝑋𝐴𝑦𝑋𝐴𝑌))
2110, 20ceqsexv 3394 . . . 4 (∃𝑦(𝑦 = 𝑌𝑋𝐴𝑦) ↔ 𝑋𝐴𝑌)
226, 19, 213bitri 286 . . 3 (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐴𝑌)
234, 5brco 5430 . . . 4 (𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ ∃𝑧(𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩))
24 ancom 448 . . . . . 6 ((𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐵𝑧))
25 vex 3354 . . . . . . . . 9 𝑧 ∈ V
2625, 5brcnv 5442 . . . . . . . 8 (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧)
2725brres 5542 . . . . . . . . 9 (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ (⟨𝑌, 𝑍⟩2nd 𝑧 ∧ ⟨𝑌, 𝑍⟩ ∈ (V × V)))
2812, 27mpbiran2 689 . . . . . . . 8 (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ ⟨𝑌, 𝑍⟩2nd 𝑧)
2910, 11br2ndeq 32009 . . . . . . . 8 (⟨𝑌, 𝑍⟩2nd 𝑧𝑧 = 𝑍)
3026, 28, 293bitri 286 . . . . . . 7 (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑧 = 𝑍)
3130anbi1i 610 . . . . . 6 ((𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ∧ 𝑋𝐵𝑧) ↔ (𝑧 = 𝑍𝑋𝐵𝑧))
3224, 31bitri 264 . . . . 5 ((𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧 = 𝑍𝑋𝐵𝑧))
3332exbii 1924 . . . 4 (∃𝑧(𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑧(𝑧 = 𝑍𝑋𝐵𝑧))
34 breq2 4791 . . . . 5 (𝑧 = 𝑍 → (𝑋𝐵𝑧𝑋𝐵𝑍))
3511, 34ceqsexv 3394 . . . 4 (∃𝑧(𝑧 = 𝑍𝑋𝐵𝑧) ↔ 𝑋𝐵𝑍)
3623, 33, 353bitri 286 . . 3 (𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐵𝑍)
3722, 36anbi12i 612 . 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 382   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  cin 3722  cop 4323   class class class wbr 4787   × cxp 5248  ccnv 5249  cres 5252  ccom 5254  1st c1st 7317  2nd c2nd 7318  ctxp 32274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fo 6036  df-fv 6038  df-1st 7319  df-2nd 7320  df-txp 32298
This theorem is referenced by:  brtxp2  32325  pprodss4v  32328  brpprod  32329  brsset  32333  brtxpsd  32338  elfuns  32359
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