Theorem opelco 5858

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⟩ ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷

Proof of Theorem opelco

Step	Hyp	Ref	Expression
1		df-br 5112	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷))
2		opelco.1	. . 3 ⊢ 𝐴 ∈ V
3		opelco.2	. . 3 ⊢ 𝐵 ∈ V
4	2, 3	brco 5857	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
5	1, 4	bitr3i 280	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400  ∃wex 1806   ∈ wcel 2149  Vcvv 3461  ⟨cop 4598   class class class wbr 5111   ∘ ccom 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5259  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-br 5112  df-opab 5176  df-co 5671

This theorem is referenced by:  dmcoss  5966  dmcossOLD  5967  dmcosseq  5969  dmcosseqOLD  5970  coiun  6259  co02  6263  coi1  6265  coass  6268  fmptco  7126  dftpos4  8241  ttrcltr  9685  fmptcof2  32943  cnvco1  36184  cnvco2  36185  txpss3v  36301  dffun10  36337  xrnss3v  38955  coiun1  44305  coxp  49531