Theorem dfatdmfcoafv2 47880

Description: Domain of a function composition, analogous to dmfco 6978. (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 47839	. . . . 5 ⊢ (𝐺 defAt 𝐴 → (𝐺''''𝐴) ∈ V)
2		opeq1 4842	. . . . . . . 8 ⊢ (𝑥 = (𝐺''''𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺''''𝐴), 𝑦⟩)
3	2	eleq1d 2854	. . . . . . 7 ⊢ (𝑥 = (𝐺''''𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
4	3	ceqsexgv 3622	. . . . . 6 ⊢ ((𝐺''''𝐴) ∈ V → (∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
5	4	bicomd 226	. . . . 5 ⊢ ((𝐺''''𝐴) ∈ V → (⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
6	1, 5	syl 18	. . . 4 ⊢ (𝐺 defAt 𝐴 → (⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
7		eqcom 2776	. . . . . . 7 ⊢ (𝑥 = (𝐺''''𝐴) ↔ (𝐺''''𝐴) = 𝑥)
8		dfatopafv2b 47872	. . . . . . . 8 ⊢ ((𝐺 defAt 𝐴 ∧ 𝑥 ∈ V) → ((𝐺''''𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
9	8	elvd 3469	. . . . . . 7 ⊢ (𝐺 defAt 𝐴 → ((𝐺''''𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
10	7, 9	bitrid 286	. . . . . 6 ⊢ (𝐺 defAt 𝐴 → (𝑥 = (𝐺''''𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
11	10	anbi1d 642	. . . . 5 ⊢ (𝐺 defAt 𝐴 → ((𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
12	11	exbidv 1948	. . . 4 ⊢ (𝐺 defAt 𝐴 → (∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
13	6, 12	bitrd 282	. . 3 ⊢ (𝐺 defAt 𝐴 → (⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
14	13	exbidv 1948	. 2 ⊢ (𝐺 defAt 𝐴 → (∃𝑦⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
15		eldm2g 5890	. . 3 ⊢ ((𝐺''''𝐴) ∈ V → ((𝐺''''𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
16	1, 15	syl 18	. 2 ⊢ (𝐺 defAt 𝐴 → ((𝐺''''𝐴) ∈ dom 𝐹 ↔ ∃𝑦⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
17		df-dfat 47745	. . 3 ⊢ (𝐺 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
18		eldm2g 5890	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺)))
19		opelco2g 5854	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
20	19	elvd 3469	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
21	20	exbidv 1948	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
22	18, 21	bitrd 282	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
23	22	adantr 485	. . 3 ⊢ ((𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
24	17, 23	sylbi 220	. 2 ⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
25	14, 16, 24	3bitr4rd 315	1 ⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463  {csn 4594  ⟨cop 4600  dom cdm 5662   ↾ cres 5664   ∘ ccom 5666  Fun wfun 6531   defAt wdfat 47742  ''''cafv2 47834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-iota 6493  df-fun 6539  df-fn 6540  df-dfat 47745  df-afv2 47835

This theorem is referenced by: (None)