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Theorem opelco 4425
 Description: Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
opelco.1 A V
opelco.2 B V
Assertion
Ref Expression
opelco (⟨A, B (𝐶𝐷) ↔ x(A𝐷x x𝐶B))
Distinct variable groups:   x,A   x,B   x,𝐶   x,𝐷

Proof of Theorem opelco
StepHypRef Expression
1 df-br 3731 . 2 (A(𝐶𝐷)B ↔ ⟨A, B (𝐶𝐷))
2 opelco.1 . . 3 A V
3 opelco.2 . . 3 B V
42, 3brco 4424 . 2 (A(𝐶𝐷)Bx(A𝐷x x𝐶B))
51, 4bitr3i 175 1 (⟨A, B (𝐶𝐷) ↔ x(A𝐷x x𝐶B))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1357   ∈ wcel 1369  Vcvv 2529  ⟨cop 3345   class class class wbr 3730   ∘ ccom 4267 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 614  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1358  ax-ie2 1359  ax-8 1371  ax-10 1372  ax-11 1373  ax-i12 1374  ax-bnd 1375  ax-4 1376  ax-14 1381  ax-17 1395  ax-i9 1399  ax-ial 1403  ax-i5r 1404  ax-ext 1998  ax-sep 3841  ax-pow 3893  ax-pr 3910 This theorem depends on definitions:  df-bi 110  df-3an 871  df-tru 1229  df-nf 1326  df-sb 1622  df-eu 1879  df-mo 1880  df-clab 2003  df-cleq 2009  df-clel 2012  df-nfc 2143  df-v 2531  df-un 2893  df-in 2895  df-ss 2902  df-pw 3328  df-sn 3348  df-pr 3349  df-op 3351  df-br 3731  df-opab 3785  df-co 4272 This theorem is referenced by:  dmcoss  4519  dmcosseq  4521  cotr  4624  coiun  4748  co02  4752  coi1  4754  coass  4757  fmptco  5246  dftpos4  5791
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