Theorem brcog 5852

Description: Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)

Assertion

Ref	Expression
brcog	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))

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

Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem brcog

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5112	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐷𝑥 ↔ 𝐴𝐷𝑥))
2		breq2 5113	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑥𝐶𝑧 ↔ 𝑥𝐶𝐵))
3	1, 2	bi2anan9 649	. . 3 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → ((𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ (𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
4	3	exbidv 1951	. 2 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐵) → (∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧) ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))
5		df-co 5670	. 2 ⊢ (𝐶 ∘ 𝐷) = {⟨𝑦, 𝑧⟩ ∣ ∃𝑥(𝑦𝐷𝑥 ∧ 𝑥𝐶𝑧)}
6	4, 5	brabga 5518	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴(𝐶 ∘ 𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥 ∧ 𝑥𝐶𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∃wex 1809   ∈ wcel 2143   class class class wbr 5109   ∘ ccom 5665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-co 5670

This theorem is referenced by:  opelco2g  5853  brcogw  5854  brco  5856  brcodir  6119  predtrss  6323  brtpos2  8224  ertr  8706  relexpindlem  15096  znleval  21704  fcoinvbr  32950  opelco3  36267  brxrn  39052  eqvreltr  39360  frege124d  44507  funressnfv  47800  dfatcolem  48012