Theorem brco 2510

Description: Binary relation on a composition.

Hypotheses

Ref	Expression
opelco.1	⊢ A ∈ V
opelco.2	⊢ B ∈ V

Assertion

Ref	Expression
brco	⊢ (A(C ∘ D)B ↔ ∃x(ADx ∧ xCB))

Distinct variable group(s):   x,A   x,B   x,C   x,D

Proof of Theorem brco

Step	Hyp	Ref	Expression
1		df-br 2063	. 2 ⊢ (A(C ∘ D)B ↔ ⟨A, B⟩ ∈ (C ∘ D))
2		opelco.1	. . 3 ⊢ A ∈ V
3		opelco.2	. . 3 ⊢ B ∈ V
4	2, 3	opelco 2509	. 2 ⊢ (⟨A, B⟩ ∈ (C ∘ D) ↔ ∃x(ADx ∧ xCB))
5	1, 4	bitr 151	1 ⊢ (A(C ∘ D)B ↔ ∃x(ADx ∧ xCB))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054   ∘ ccom 2414

This theorem is referenced by:  funco 2696

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-co 2427

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