Theorem opelco 2509

Description: Ordered pair membership in a composition.

Hypotheses

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

Assertion

Ref	Expression
opelco	⊢ (⟨A, B⟩ ∈ (C ∘ D) ↔ ∃x(ADx ∧ xCB))

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

Proof of Theorem opelco

Step	Hyp	Ref	Expression
1		df-co 2427	. . 3 ⊢ (C ∘ D) = {⟨y, z⟩∣∃x(yDx ∧ xCz)}
2	1	eleq2i 1153	. 2 ⊢ (⟨A, B⟩ ∈ (C ∘ D) ↔ ⟨A, B⟩ ∈ {⟨y, z⟩∣∃x(yDx ∧ xCz)})
3		opelco.1	. . 3 ⊢ A ∈ V
4		opelco.2	. . 3 ⊢ B ∈ V
5		breq1 2065	. . . . 5 ⊢ (y = A → (yDx ↔ ADx))
6	5	anbi1d 469	. . . 4 ⊢ (y = A → ((yDx ∧ xCz) ↔ (ADx ∧ xCz)))
7	6	biexdv 936	. . 3 ⊢ (y = A → (∃x(yDx ∧ xCz) ↔ ∃x(ADx ∧ xCz)))
8		breq2 2066	. . . . 5 ⊢ (z = B → (xCz ↔ xCB))
9	8	anbi2d 468	. . . 4 ⊢ (z = B → ((ADx ∧ xCz) ↔ (ADx ∧ xCB)))
10	9	biexdv 936	. . 3 ⊢ (z = B → (∃x(ADx ∧ xCz) ↔ ∃x(ADx ∧ xCB)))
11	3, 4, 7, 10	opelopab 2117	. 2 ⊢ (⟨A, B⟩ ∈ {⟨y, z⟩∣∃x(yDx ∧ xCz)} ↔ ∃x(ADx ∧ xCB))
12	2, 11	bitr 151	1 ⊢ (⟨A, B⟩ ∈ (C ∘ D) ↔ ∃x(ADx ∧ xCB))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054  {copab 2055   ∘ ccom 2414

This theorem is referenced by:  brco 2510  opelcog 2511  cnvco 2520  dmco 2570  dmcosseq 2572  cores 2659  co02 2663  coi1 2665  coass 2667  dmco2 2673

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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