Theorem dfatdmfcoafv2 47717

Description: Domain of a function composition, analogous to dmfco 6923. (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 47676	. . . . 5 ⊢ (𝐺 defAt 𝐴 → (𝐺''''𝐴) ∈ V)
2		opeq1 4804	. . . . . . . 8 ⊢ (𝑥 = (𝐺''''𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺''''𝐴), 𝑦⟩)
3	2	eleq1d 2824	. . . . . . 7 ⊢ (𝑥 = (𝐺''''𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
4	3	ceqsexgv 3592	. . . . . 6 ⊢ ((𝐺''''𝐴) ∈ V → (∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
5	4	bicomd 224	. . . . 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 2746	. . . . . . 7 ⊢ (𝑥 = (𝐺''''𝐴) ↔ (𝐺''''𝐴) = 𝑥)
8		dfatopafv2b 47709	. . . . . . . 8 ⊢ ((𝐺 defAt 𝐴 ∧ 𝑥 ∈ V) → ((𝐺''''𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
9	8	elvd 3437	. . . . . . 7 ⊢ (𝐺 defAt 𝐴 → ((𝐺''''𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
10	7, 9	bitrid 284	. . . . . 6 ⊢ (𝐺 defAt 𝐴 → (𝑥 = (𝐺''''𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
11	10	anbi1d 637	. . . . 5 ⊢ (𝐺 defAt 𝐴 → ((𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
12	11	exbidv 1928	. . . 4 ⊢ (𝐺 defAt 𝐴 → (∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
13	6, 12	bitrd 280	. . 3 ⊢ (𝐺 defAt 𝐴 → (⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
14	13	exbidv 1928	. 2 ⊢ (𝐺 defAt 𝐴 → (∃𝑦⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
15		eldm2g 5841	. . 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 47582	. . 3 ⊢ (𝐺 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
18		eldm2g 5841	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺)))
19		opelco2g 5809	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
20	19	elvd 3437	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
21	20	exbidv 1928	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
22	18, 21	bitrd 280	. . . 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 218	. 2 ⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
25	14, 16, 24	3bitr4rd 313	1 ⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3431  {csn 4555  ⟨cop 4561  dom cdm 5618   ↾ cres 5620   ∘ ccom 5622  Fun wfun 6479   defAt wdfat 47579  ''''cafv2 47671

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-res 5630  df-iota 6441  df-fun 6487  df-fn 6488  df-dfat 47582  df-afv2 47672

This theorem is referenced by: (None)