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Theorem brcog 5880
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 5151 . . . 4 (𝑦 = 𝐴 → (𝑦𝐷𝑥𝐴𝐷𝑥))
2 breq2 5152 . . . 4 (𝑧 = 𝐵 → (𝑥𝐶𝑧𝑥𝐶𝐵))
31, 2bi2anan9 638 . . 3 ((𝑦 = 𝐴𝑧 = 𝐵) → ((𝑦𝐷𝑥𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥𝑥𝐶𝐵)))
43exbidv 1919 . 2 ((𝑦 = 𝐴𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
5 df-co 5698 . 2 (𝐶𝐷) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐷𝑥𝑥𝐶𝑧)}
64, 5brabga 5544 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106   class class class wbr 5148  ccom 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-co 5698
This theorem is referenced by:  opelco2g  5881  brcogw  5882  brco  5884  brcodir  6142  predtrss  6345  brtpos2  8256  ertr  8759  relexpindlem  15099  znleval  21591  fcoinvbr  32625  opelco3  35756  brxrn  38356  eqvreltr  38589  frege124d  43751  funressnfv  46993  dfatcolem  47205
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