Theorem dfatdmfcoafv2 47614

Description: Domain of a function composition, analogous to dmfco 6938. (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 47573	. . . . 5 ⊢ (𝐺 defAt 𝐴 → (𝐺''''𝐴) ∈ V)
2		opeq1 4831	. . . . . . . 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 3610	. . . . . 6 ⊢ ((𝐺''''𝐴) ∈ V → (∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹))
5	4	bicomd 223	. . . . 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 2744	. . . . . . 7 ⊢ (𝑥 = (𝐺''''𝐴) ↔ (𝐺''''𝐴) = 𝑥)
8		dfatopafv2b 47606	. . . . . . . 8 ⊢ ((𝐺 defAt 𝐴 ∧ 𝑥 ∈ V) → ((𝐺''''𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
9	8	elvd 3448	. . . . . . 7 ⊢ (𝐺 defAt 𝐴 → ((𝐺''''𝐴) = 𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
10	7, 9	bitrid 283	. . . . . 6 ⊢ (𝐺 defAt 𝐴 → (𝑥 = (𝐺''''𝐴) ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐺))
11	10	anbi1d 632	. . . . 5 ⊢ (𝐺 defAt 𝐴 → ((𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
12	11	exbidv 1923	. . . 4 ⊢ (𝐺 defAt 𝐴 → (∃𝑥(𝑥 = (𝐺''''𝐴) ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
13	6, 12	bitrd 279	. . 3 ⊢ (𝐺 defAt 𝐴 → (⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
14	13	exbidv 1923	. 2 ⊢ (𝐺 defAt 𝐴 → (∃𝑦⟨(𝐺''''𝐴), 𝑦⟩ ∈ 𝐹 ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
15		eldm2g 5856	. . 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 47479	. . 3 ⊢ (𝐺 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})))
18		eldm2g 5856	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺)))
19		opelco2g 5824	. . . . . . 7 ⊢ ((𝐴 ∈ dom 𝐺 ∧ 𝑦 ∈ V) → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
20	19	elvd 3448	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐺 → (⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
21	20	exbidv 1923	. . . . 5 ⊢ (𝐴 ∈ dom 𝐺 → (∃𝑦⟨𝐴, 𝑦⟩ ∈ (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
22	18, 21	bitrd 279	. . . 4 ⊢ (𝐴 ∈ dom 𝐺 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
23	22	adantr 480	. . 3 ⊢ ((𝐴 ∈ dom 𝐺 ∧ Fun (𝐺 ↾ {𝐴})) → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
24	17, 23	sylbi 217	. 2 ⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ ∃𝑦∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐺 ∧ ⟨𝑥, 𝑦⟩ ∈ 𝐹)))
25	14, 16, 24	3bitr4rd 312	1 ⊢ (𝐺 defAt 𝐴 → (𝐴 ∈ dom (𝐹 ∘ 𝐺) ↔ (𝐺''''𝐴) ∈ dom 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442  {csn 4582  ⟨cop 4588  dom cdm 5632   ↾ cres 5634   ∘ ccom 5636  Fun wfun 6494   defAt wdfat 47476  ''''cafv2 47568

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-dfat 47479  df-afv2 47569

This theorem is referenced by: (None)