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Theorem opelco2g 4426
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 4425 . 2 ((A 𝑉 B 𝑊) → (A(𝐶𝐷)Bx(A𝐷x x𝐶B)))
2 df-br 3735 . 2 (A(𝐶𝐷)B ↔ ⟨A, B (𝐶𝐷))
3 df-br 3735 . . . 4 (A𝐷x ↔ ⟨A, x 𝐷)
4 df-br 3735 . . . 4 (x𝐶B ↔ ⟨x, B 𝐶)
53, 4anbi12i 436 . . 3 ((A𝐷x x𝐶B) ↔ (⟨A, x 𝐷 x, B 𝐶))
65exbii 1474 . 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 1358   wcel 1370  cop 3349   class class class wbr 3734  ccom 4272
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-co 4277
This theorem is referenced by:  dfco2  4743  dmfco  5162
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