Theorem dfatdmfcoafv2 45476

Description: Domain of a function composition, analogous to dmfco 6937. (Contributed by AV, 7-Sep-2022.)

Assertion

Ref	Expression
dfatdmfcoafv2	⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))

Proof of Theorem dfatdmfcoafv2

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

Step	Hyp	Ref	Expression
1		dfatafv2ex 45435	. . . . 5 ⊢ (𝐺 defAt 𝐴 → (𝐺''''𝐴) ∈ V)
2		opeq1 4830	. . . . . . . 8 ⊢ (𝑥 = (𝐺''''𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺''''𝐴), 𝑦⟩)
3	2	eleq1d 2822	. . . . . . 7 ⊢ (𝑥 = (𝐺''''𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
4	3	ceqsexgv 3604	. . . . . 6 ⊢ ((𝐺''''𝐴) ∈ V → (∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
5	4	bicomd 222	. . . . 5 ⊢ ((𝐺''''𝐴) ∈ V → (⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
6	1, 5	syl 17	. . . 4 ⊢ (𝐺 defAt 𝐴 → (⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7		eqcom 2743	. . . . . . 7 ⊢ (𝑥 = (𝐺''''𝐴) ↔ (𝐺''''𝐴) = 𝑥)
8		dfatopafv2b 45468	. . . . . . . 8 ⊢ ((𝐺 defAt 𝐴 ∧ 𝑥 ∈ V) → ((𝐺''''𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
9	8	elvd 3452	. . . . . . 7 ⊢ (𝐺 defAt 𝐴 → ((𝐺''''𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
10	7, 9	bitrid 282	. . . . . 6 ⊢ (𝐺 defAt 𝐴 → (𝑥 = (𝐺''''𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
11	10	anbi1d 630	. . . . 5 ⊢ (𝐺 defAt 𝐴 → ((𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
12	11	exbidv 1924	. . . 4 ⊢ (𝐺 defAt 𝐴 → (∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
13	6, 12	bitrd 278	. . 3 ⊢ (𝐺 defAt 𝐴 → (⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
14	13	exbidv 1924	. 2 ⊢ (𝐺 defAt 𝐴 → (∃𝑦⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
15		eldm2g 5855	. . 3 ⊢ ((𝐺''''𝐴) ∈ V → ((𝐺''''𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
16	1, 15	syl 17	. 2 ⊢ (𝐺 defAt 𝐴 → ((𝐺''''𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
17		df-dfat 45341	. . 3 ⊢ (𝐺 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
18		eldm2g 5855	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺)))
19		opelco2g 5823	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
20	19	elvd 3452	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
21	20	exbidv 1924	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
22	18, 21	bitrd 278	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
23	22	adantr 481	. . 3 ⊢ ((𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
24	17, 23	sylbi 216	. 2 ⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
25	14, 16, 24	3bitr4rd 311	1 ⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3445  {csn 4586  ⟨cop 4592  dom cdm 5633   ↾ cres 5635   ∘ ccom 5637  Fun wfun 6490   defAt wdfat 45338  ''''cafv2 45430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-res 5645  df-iota 6448  df-fun 6498  df-fn 6499  df-dfat 45341  df-afv2 45431

This theorem is referenced by: (None)