Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brtxp Structured version   Visualization version   GIF version

Theorem brtxp 33238
Description: Characterize a ternary relation over a tail Cartesian product. Together with txpss3v 33236, 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 33212 . . 3 (𝐴𝐵) = (((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))
21breqi 5063 . 2 (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ 𝑋(((1st ↾ (V × V)) ∘ 𝐴) ∩ ((2nd ↾ (V × V)) ∘ 𝐵))⟨𝑌, 𝑍⟩)
3 brin 5109 . 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 5347 . . . . 5 𝑌, 𝑍⟩ ∈ V
64, 5brco 5734 . . . 4 (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ ∃𝑦(𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩))
7 vex 3495 . . . . . . . 8 𝑦 ∈ V
87, 5brcnv 5746 . . . . . . 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 5587 . . . . . . . 8 𝑌, 𝑍⟩ ∈ (V × V)
127brresi 5855 . . . . . . . 8 (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ (⟨𝑌, 𝑍⟩ ∈ (V × V) ∧ ⟨𝑌, 𝑍⟩1st 𝑦))
1311, 12mpbiran 705 . . . . . . 7 (⟨𝑌, 𝑍⟩(1st ↾ (V × V))𝑦 ↔ ⟨𝑌, 𝑍⟩1st 𝑦)
149, 10br1steq 32911 . . . . . . 7 (⟨𝑌, 𝑍⟩1st 𝑦𝑦 = 𝑌)
158, 13, 143bitri 298 . . . . . 6 (𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑦 = 𝑌)
1615anbi1ci 625 . . . . 5 ((𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑦 = 𝑌𝑋𝐴𝑦))
1716exbii 1839 . . . 4 (∃𝑦(𝑋𝐴𝑦𝑦(1st ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑦(𝑦 = 𝑌𝑋𝐴𝑦))
18 breq2 5061 . . . . 5 (𝑦 = 𝑌 → (𝑋𝐴𝑦𝑋𝐴𝑌))
199, 18ceqsexv 3539 . . . 4 (∃𝑦(𝑦 = 𝑌𝑋𝐴𝑦) ↔ 𝑋𝐴𝑌)
206, 17, 193bitri 298 . . 3 (𝑋((1st ↾ (V × V)) ∘ 𝐴)⟨𝑌, 𝑍⟩ ↔ 𝑋𝐴𝑌)
214, 5brco 5734 . . . 4 (𝑋((2nd ↾ (V × V)) ∘ 𝐵)⟨𝑌, 𝑍⟩ ↔ ∃𝑧(𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩))
22 vex 3495 . . . . . . . 8 𝑧 ∈ V
2322, 5brcnv 5746 . . . . . . 7 (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ ⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧)
2422brresi 5855 . . . . . . . 8 (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ (⟨𝑌, 𝑍⟩ ∈ (V × V) ∧ ⟨𝑌, 𝑍⟩2nd 𝑧))
2511, 24mpbiran 705 . . . . . . 7 (⟨𝑌, 𝑍⟩(2nd ↾ (V × V))𝑧 ↔ ⟨𝑌, 𝑍⟩2nd 𝑧)
269, 10br2ndeq 32912 . . . . . . 7 (⟨𝑌, 𝑍⟩2nd 𝑧𝑧 = 𝑍)
2723, 25, 263bitri 298 . . . . . 6 (𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩ ↔ 𝑧 = 𝑍)
2827anbi1ci 625 . . . . 5 ((𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ (𝑧 = 𝑍𝑋𝐵𝑧))
2928exbii 1839 . . . 4 (∃𝑧(𝑋𝐵𝑧𝑧(2nd ↾ (V × V))⟨𝑌, 𝑍⟩) ↔ ∃𝑧(𝑧 = 𝑍𝑋𝐵𝑧))
30 breq2 5061 . . . . 5 (𝑧 = 𝑍 → (𝑋𝐵𝑧𝑋𝐵𝑍))
3110, 30ceqsexv 3539 . . . 4 (∃𝑧(𝑧 = 𝑍𝑋𝐵𝑧) ↔ 𝑋𝐵𝑍)
3221, 29, 313bitri 298 . . 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 298 1 (𝑋(𝐴𝐵)⟨𝑌, 𝑍⟩ ↔ (𝑋𝐴𝑌𝑋𝐵𝑍))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  cin 3932  cop 4563   class class class wbr 5057   × cxp 5546  ccnv 5547  cres 5550  ccom 5552  1st c1st 7676  2nd c2nd 7677  ctxp 33188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678  df-2nd 7679  df-txp 33212
This theorem is referenced by:  brtxp2  33239  pprodss4v  33242  brpprod  33243  brsset  33247  brtxpsd  33252  elfuns  33273
  Copyright terms: Public domain W3C validator