Theorem dmfco 6759

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 5770	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺)))
2		opelco2g 5740	. . . . . 6 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
3	2	elvd 3502	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
4	3	exbidv 1922	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
5	1, 4	bitrd 281	. . 3 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
6	5	adantl 484	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7		fvex 6685	. . . 4 ⊢ (𝐺‘𝐴) ∈ V
8	7	eldm2 5772	. . 3 ⊢ ((𝐺‘𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹)
9		opeq1 4805	. . . . . . 7 ⊢ (𝑥 = (𝐺‘𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺‘𝐴), 𝑦⟩)
10	9	eleq1d 2899	. . . . . 6 ⊢ (𝑥 = (𝐺‘𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹))
11	7, 10	ceqsexv 3543	. . . . 5 ⊢ (∃𝑥(𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹)
12		eqcom 2830	. . . . . . . 8 ⊢ (𝑥 = (𝐺‘𝐴) ↔ (𝐺‘𝐴) = 𝑥)
13		funopfvb 6723	. . . . . . . 8 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐺‘𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
14	12, 13	syl5bb 285	. . . . . . 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 1922	. . . . 5 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (∃𝑥(𝑥 = (𝐺‘𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
17	11, 16	syl5bbr 287	. . . 4 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
18	17	exbidv 1922	. . 3 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (∃𝑦⟨(𝐺‘𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
19	8, 18	syl5bb 285	. 2 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → ((𝐺‘𝐴) ∈ dom 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
20	6, 19	bitr4d 284	1 ⊢ ((Fun 𝐺 ∧ 𝐴 ∈ dom 𝐺) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺‘𝐴) ∈ dom 𝐹))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3496  ⟨cop 4575  dom cdm 5557   ∘ ccom 5561  Fun wfun 6351  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fn 6360  df-fv 6365

This theorem is referenced by:  hoicvr  42837  funressnfv  43285  dmfcoafv  43381  afvco2  43382