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Theorem brcog 4445
Description: Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
Assertion
Ref Expression
brcog ((A 𝑉 B 𝑊) → (A(𝐶𝐷)Bx(A𝐷x x𝐶B)))
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 3758 . . . 4 (y = A → (y𝐷xA𝐷x))
2 breq2 3759 . . . 4 (z = B → (x𝐶zx𝐶B))
31, 2bi2anan9 538 . . 3 ((y = A z = B) → ((y𝐷x x𝐶z) ↔ (A𝐷x x𝐶B)))
43exbidv 1703 . 2 ((y = A z = B) → (x(y𝐷x x𝐶z) ↔ x(A𝐷x x𝐶B)))
5 df-co 4297 . 2 (𝐶𝐷) = {⟨y, z⟩ ∣ x(y𝐷x x𝐶z)}
64, 5brabga 3992 1 ((A 𝑉 B 𝑊) → (A(𝐶𝐷)Bx(A𝐷x x𝐶B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390   class class class wbr 3755  ccom 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-co 4297
This theorem is referenced by:  opelco2g  4446  brcogw  4447  brco  4449  brcodir  4655  foeqcnvco  5373  brtpos2  5807  ertr  6057
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