Theorem dmfco 6960

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 5866	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺)))
2		opelco2g 5834	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
3	2	elvd 3456	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
4	3	exbidv 1921	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
5	1, 4	bitrd 279	. . 3 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
6	5	adantl 481	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7		fvex 6874	. . . 4 ⊢ (𝐺‘𝐴) ∈ V
8	7	eldm2 5868	. . 3 ⊢ ((𝐺‘𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹)
9		opeq1 4840	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺‘𝐴), 𝑦⟩)
10	9	eleq1d 2814	. . . . . 6 ⊢ (𝑥 = (𝐺‘𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹))
11	7, 10	ceqsexv 3501	. . . . 5 ⊢ (∃𝑥(𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹)
12		eqcom 2737	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝐴) ↔ (𝐺‘𝐴) = 𝑥)
13		funopfvb 6918	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐺‘𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
14	12, 13	bitrid 283	. . . . . . 7 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝑥 = (𝐺‘𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
15	14	anbi1d 631	. . . . . 6 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
16	15	exbidv 1921	. . . . 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 1921	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (∃𝑦⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
19	8, 18	bitrid 283	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐺‘𝐴) ∈ dom 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
20	6, 19	bitr4d 282	1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝐴) ∈ dom 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450  ⟨cop 4598  dom cdm 5641   ∘ ccom 5645  Fun wfun 6508  ‘cfv 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fn 6517  df-fv 6522

This theorem is referenced by:  hoicvr  46553  funressnfv  47048  dmfcoafv  47180  afvco2  47181