Theorem dfatdmfcoafv2 48323

Description: Domain of a function composition, analogous to dmfco 6981. (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 48282	. . . . 5 ⊢ (𝐺 defAt 𝐴 → (𝐺''''𝐴) ∈ V)
2		opeq1 4833	. . . . . . . 8 ⊢ (𝑥 = (𝐺''''𝐴) → ⟨𝑥, 𝑦⟩ = ⟨(𝐺''''𝐴), 𝑦⟩)
3	2	eleq1d 2846	. . . . . . 7 ⊢ (𝑥 = (𝐺''''𝐴) → (⟨𝑥, 𝑦⟩ ∈ 𝐹 ↔ ⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
4	3	ceqsexgv 3608	. . . . . 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 2768	. . . . . . 7 ⊢ (𝑥 = (𝐺''''𝐴) ↔ (𝐺''''𝐴) = 𝑥)
8		dfatopafv2b 48315	. . . . . . . 8 ⊢ ((𝐺 defAt 𝐴 ∧ 𝑥 ∈ V) → ((𝐺''''𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
9	8	elvd 3457	. . . . . . 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 643	. . . . 5 ⊢ (𝐺 defAt 𝐴 → ((𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
12	11	exbidv 1954	. . . 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 1954	. 2 ⊢ (𝐺 defAt 𝐴 → (∃𝑦⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
15		eldm2g 5881	. . 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 48188	. . 3 ⊢ (𝐺 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
18		eldm2g 5881	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺)))
19		opelco2g 5845	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
20	19	elvd 3457	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
21	20	exbidv 1954	. . . . 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 486	. . 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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  Vcvv 3451  {csn 4584  ⟨cop 4590  dom cdm 5651   ↾ cres 5653   ∘ ccom 5655  Fun wfun 6532   defAt wdfat 48185  ''''cafv2 48277

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-res 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-dfat 48188  df-afv2 48278

This theorem is used by: (None)