MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brcog Structured version   Visualization version   GIF version

Theorem brcog 5534
Description: Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
Assertion
Ref Expression
brcog ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem brcog
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4889 . . . 4 (𝑦 = 𝐴 → (𝑦𝐷𝑥𝐴𝐷𝑥))
2 breq2 4890 . . . 4 (𝑧 = 𝐵 → (𝑥𝐶𝑧𝑥𝐶𝐵))
31, 2bi2anan9 629 . . 3 ((𝑦 = 𝐴𝑧 = 𝐵) → ((𝑦𝐷𝑥𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥𝑥𝐶𝐵)))
43exbidv 1964 . 2 ((𝑦 = 𝐴𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
5 df-co 5364 . 2 (𝐶𝐷) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐷𝑥𝑥𝐶𝑧)}
64, 5brabga 5226 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wex 1823  wcel 2107   class class class wbr 4886  ccom 5359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-co 5364
This theorem is referenced by:  opelco2g  5535  brcogw  5536  brco  5538  brcodir  5770  brtpos2  7640  ertr  8041  relexpindlem  14210  znleval  20298  fcoinvbr  29982  opelco3  32266  brxrn  34764  eqvreltr  34977  frege124d  39010  funressnfv  42107  dfatcolem  42296
  Copyright terms: Public domain W3C validator