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Theorem brtxp 35607
Description: Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 35605, this completely defines membership in a tail cross. (Contributed by Scott Fenton, 31-Mar-2012.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
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 35581 . . 3 (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
21breqi 5155 . 2 (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋(((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩)
3 brin 5201 . 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 5466 . . . . 5 𝑌, 𝑍⟩ ∈ V
64, 5brco 5873 . . . 4 (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ ∃𝑦(𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩))
7 vex 3465 . . . . . . . 8 𝑦 ∈ V
87, 5brcnv 5885 . . . . . . 7 (𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦)
9 brtxp.2 . . . . . . . . 9 𝑌 ∈ V
10 brtxp.3 . . . . . . . . 9 𝑍 ∈ V
119, 10opelvv 5718 . . . . . . . 8 𝑌, 𝑍⟩ ∈ (V × V)
127brresi 5994 . . . . . . . 8 (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ (⟨𝑌, 𝑍⟩ ∈ (V × V) ∧ ⟨𝑌, 𝑍⟩1st 𝑦))
1311, 12mpbiran 707 . . . . . . 7 (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ ⟨𝑌, 𝑍⟩1st 𝑦)
149, 10br1steq 35497 . . . . . . 7 (⟨𝑌, 𝑍⟩1st 𝑦𝑦 = 𝑌)
158, 13, 143bitri 296 . . . . . 6 (𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑦 = 𝑌)
1615anbi1ci 624 . . . . 5 ((𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦 = 𝑌𝑋𝐴𝑦))
1716exbii 1842 . . . 4 (∃𝑦(𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑦(𝑦 = 𝑌𝑋𝐴𝑦))
18 breq2 5153 . . . . 5 (𝑦 = 𝑌 → (𝑋𝐴𝑦𝑋𝐴𝑌))
199, 18ceqsexv 3514 . . . 4 (∃𝑦(𝑦 = 𝑌𝑋𝐴𝑦) ↔ 𝑋𝐴𝑌)
206, 17, 193bitri 296 . . 3 (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐴𝑌)
214, 5brco 5873 . . . 4 (𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ ∃𝑧(𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩))
22 vex 3465 . . . . . . . 8 𝑧 ∈ V
2322, 5brcnv 5885 . . . . . . 7 (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧)
2422brresi 5994 . . . . . . . 8 (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ (⟨𝑌, 𝑍⟩ ∈ (V × V) ∧ ⟨𝑌, 𝑍⟩2nd 𝑧))
2511, 24mpbiran 707 . . . . . . 7 (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ ⟨𝑌, 𝑍⟩2nd 𝑧)
269, 10br2ndeq 35498 . . . . . . 7 (⟨𝑌, 𝑍⟩2nd 𝑧𝑧 = 𝑍)
2723, 25, 263bitri 296 . . . . . 6 (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑧 = 𝑍)
2827anbi1ci 624 . . . . 5 ((𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧 = 𝑍𝑋𝐵𝑧))
2928exbii 1842 . . . 4 (∃𝑧(𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑧(𝑧 = 𝑍𝑋𝐵𝑧))
30 breq2 5153 . . . . 5 (𝑧 = 𝑍 → (𝑋𝐵𝑧𝑋𝐵𝑍))
3110, 30ceqsexv 3514 . . . 4 (∃𝑧(𝑧 = 𝑍𝑋𝐵𝑧) ↔ 𝑋𝐵𝑍)
3221, 29, 313bitri 296 . . 3 (𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐵𝑍)
3320, 32anbi12i 626 . 2 ((𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ∧ 𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩) ↔ (𝑋𝐴𝑌𝑋𝐵𝑍))
342, 3, 333bitri 296 1 (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌𝑋𝐵𝑍))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  Vcvv 3461  cin 3943  cop 4636   class class class wbr 5149   × cxp 5676  ccnv 5677  cres 5680  ccom 5682  1st c1st 7992  2nd c2nd 7993  ctxp 35557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fo 6555  df-fv 6557  df-1st 7994  df-2nd 7995  df-txp 35581
This theorem is referenced by:  brtxp2  35608  pprodss4v  35611  brpprod  35612  brsset  35616  brtxpsd  35621  elfuns  35642
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