Theorem opelco 4801

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 4006	. 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 4800	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))
5	1, 4	bitr3i 186	1 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1492   ∈ wcel 2148  Vcvv 2739  ⟨cop 3597   class class class wbr 4005   ∘ ccom 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-co 4637

This theorem is referenced by:  dmcoss  4898  dmcosseq  4900  cotr  5012  coiun  5140  co02  5144  coi1  5146  coass  5149  fmptco  5684  dftpos4  6266