Theorem opelco2g 5874

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⟩ ∈ (𝐶 ∘ 𝐷) ↔ ∃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

Step	Hyp	Ref	Expression
1		brcog 5873	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
2		df-br 5153	. 2 ⊢ (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷))
3		df-br 5153	. . . 4 ⊢ (𝐴𝐷𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐷)
4		df-br 5153	. . . 4 ⊢ (𝑥𝐶𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ 𝐶)
5	3, 4	anbi12i 626	. . 3 ⊢ ((𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶))
6	5	exbii 1842	. 2 ⊢ (∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶))
7	1, 2, 6	3bitr3g 312	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (⟨𝐴, 𝐵⟩ ∈ (𝐶 ∘ 𝐷) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394  ∃wex 1773   ∈ wcel 2098  ⟨cop 4638   class class class wbr 5152   ∘ ccom 5686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-co 5691

This theorem is referenced by:  dfco2  6254  dmfco  6999  dfatdmfcoafv2  46663