Theorem dmfco 6864

Description: Domains of a function composition. (Contributed by NM, 27-Jan-1997.)

Assertion

Ref	Expression
dmfco	⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝐴) ∈ dom 𝐹))

Proof of Theorem dmfco

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

Step	Hyp	Ref	Expression
1		eldm2g 5808	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺)))
2		opelco2g 5776	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
3	2	elvd 3439	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
4	3	exbidv 1924	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
5	1, 4	bitrd 278	. . 3 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
6	5	adantl 482	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7		fvex 6787	. . . 4 ⊢ (𝐺‘𝐴) ∈ V
8	7	eldm2 5810	. . 3 ⊢ ((𝐺‘𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹)
9		opeq1 4804	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺‘𝐴), 𝑦⟩)
10	9	eleq1d 2823	. . . . . 6 ⊢ (𝑥 = (𝐺‘𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹))
11	7, 10	ceqsexv 3479	. . . . 5 ⊢ (∃𝑥(𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹)
12		eqcom 2745	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝐴) ↔ (𝐺‘𝐴) = 𝑥)
13		funopfvb 6825	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐺‘𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
14	12, 13	bitrid 282	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝑥 = (𝐺‘𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
15	14	anbi1d 630	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
16	15	exbidv 1924	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (∃𝑥(𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
17	11, 16	bitr3id 285	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
18	17	exbidv 1924	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (∃𝑦⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
19	8, 18	bitrid 282	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐺‘𝐴) ∈ dom 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
20	6, 19	bitr4d 281	1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝐴) ∈ dom 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432  ⟨cop 4567  dom cdm 5589   ∘ ccom 5593  Fun wfun 6427  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  hoicvr  44086  funressnfv  44537  dmfcoafv  44667  afvco2  44668