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Theorem brcog 4550
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 3808 . . . 4 (𝑦 = 𝐴 → (𝑦𝐷𝑥𝐴𝐷𝑥))
2 breq2 3809 . . . 4 (𝑧 = 𝐵 → (𝑥𝐶𝑧𝑥𝐶𝐵))
31, 2bi2anan9 571 . . 3 ((𝑦 = 𝐴𝑧 = 𝐵) → ((𝑦𝐷𝑥𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥𝑥𝐶𝐵)))
43exbidv 1748 . 2 ((𝑦 = 𝐴𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
5 df-co 4400 . 2 (𝐶𝐷) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐷𝑥𝑥𝐶𝑧)}
64, 5brabga 4047 1 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1285  wex 1422  wcel 1434   class class class wbr 3805  ccom 4395
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-co 4400
This theorem is referenced by:  opelco2g  4551  brcogw  4552  brco  4554  brcodir  4762  foeqcnvco  5481  brtpos2  5920  ertr  6208
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