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Theorem opelco2g 4446
Description: Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
Assertion
Ref Expression
opelco2g ((A 𝑉 B 𝑊) → (⟨A, B (𝐶𝐷) ↔ x(⟨A, x 𝐷 x, B 𝐶)))
Distinct variable groups:   x,A   x,B   x,𝐶   x,𝐷
Allowed substitution hints:   𝑉(x)   𝑊(x)

Proof of Theorem opelco2g
StepHypRef Expression
1 brcog 4445 . 2 ((A 𝑉 B 𝑊) → (A(𝐶𝐷)Bx(A𝐷x x𝐶B)))
2 df-br 3756 . 2 (A(𝐶𝐷)B ↔ ⟨A, B (𝐶𝐷))
3 df-br 3756 . . . 4 (A𝐷x ↔ ⟨A, x 𝐷)
4 df-br 3756 . . . 4 (x𝐶B ↔ ⟨x, B 𝐶)
53, 4anbi12i 433 . . 3 ((A𝐷x x𝐶B) ↔ (⟨A, x 𝐷 x, B 𝐶))
65exbii 1493 . 2 (x(A𝐷x x𝐶B) ↔ x(⟨A, x 𝐷 x, B 𝐶))
71, 2, 63bitr3g 211 1 ((A 𝑉 B 𝑊) → (⟨A, B (𝐶𝐷) ↔ x(⟨A, x 𝐷 x, B 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wex 1378   wcel 1390  cop 3370   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:  dfco2  4763  dmfco  5184
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