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Theorem brcog 4713
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 3939 . . . 4 (𝑦 = 𝐴 → (𝑦𝐷𝑥𝐴𝐷𝑥))
2 breq2 3940 . . . 4 (𝑧 = 𝐵 → (𝑥𝐶𝑧𝑥𝐶𝐵))
31, 2bi2anan9 596 . . 3 ((𝑦 = 𝐴𝑧 = 𝐵) → ((𝑦𝐷𝑥𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥𝑥𝐶𝐵)))
43exbidv 1798 . 2 ((𝑦 = 𝐴𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
5 df-co 4555 . 2 (𝐶𝐷) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐷𝑥𝑥𝐶𝑧)}
64, 5brabga 4193 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1332  wex 1469  wcel 1481   class class class wbr 3936  ccom 4550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-co 4555
This theorem is referenced by:  opelco2g  4714  brcogw  4715  brco  4717  brcodir  4933  foeqcnvco  5698  brtpos2  6155  ertr  6451
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