Theorem opelco 5896

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 5167	. 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 5895	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
5	1, 4	bitr3i 277	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  Vcvv 3488  ⟨cop 4654   class class class wbr 5166   ∘ ccom 5704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-co 5709

This theorem is referenced by:  dmcoss  5997  dmcosseq  5999  dmcosseqOLD  6000  cotrgOLDOLD  6141  coiun  6287  co02  6291  coi1  6293  coass  6296  fmptco  7163  dftpos4  8286  ttrcltr  9785  fmptcof2  32675  cnvco1  35721  cnvco2  35722  txpss3v  35842  dffun10  35878  xrnss3v  38328  coiun1  43614