Theorem nninffeq 17229

Description: Equality of two functions on ℕ_∞ which agree at every integer and at the point at infinity. From an online post by Martin Escardo. Remark: the last two hypotheses can be grouped into one, ⊢ (𝜑 → ∀𝑛 ∈ suc ω...). (Contributed by Jim Kingdon, 4-Aug-2023.)

Hypotheses

Ref	Expression
nninffeq.f	⊢ (𝜑 → 𝐹:ℕ∞⟶ℕ0)
nninffeq.g	⊢ (𝜑 → 𝐺:ℕ∞⟶ℕ0)
nninffeq.oo	⊢ (𝜑 → (𝐹‘(𝑥 ∈ ω ↦ 1o)) = (𝐺‘(𝑥 ∈ ω ↦ 1o)))
nninffeq.n	⊢ (𝜑 → ∀𝑛 ∈ ω (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))

Assertion

Ref	Expression
nninffeq	⊢ (𝜑 → 𝐹 = 𝐺)

Distinct variable groups:   𝑖,𝐹,𝑛,𝑥   𝑖,𝐺,𝑛,𝑥   𝜑,𝑖,𝑛,𝑥

Proof of Theorem nninffeq

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nninffeq.f	. . 3 ⊢ (𝜑 → 𝐹:ℕ∞⟶ℕ0)
2	1	ffnd 5534	. 2 ⊢ (𝜑 → 𝐹 Fn ℕ∞)
3		nninffeq.g	. . 3 ⊢ (𝜑 → 𝐺:ℕ∞⟶ℕ0)
4	3	ffnd 5534	. 2 ⊢ (𝜑 → 𝐺 Fn ℕ∞)
5		eqid 2238	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅)) = (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))
6		fveq2 5695	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
7		fveq2 5695	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
8	6, 7	eqeq12d 2253	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝑧) = (𝐺‘𝑧)))
9	8	ifbid 3662	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅) = if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅))
10		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → 𝑧 ∈ ℕ∞)
11		1onn 6793	. . . . . . . . . 10 ⊢ 1o ∈ ω
12	11	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → 1o ∈ ω)
13		peano1 4741	. . . . . . . . . 10 ⊢ ∅ ∈ ω
14	13	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ∅ ∈ ω)
15	1	ffvelcdmda 5843	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐹‘𝑧) ∈ ℕ0)
16	15	nn0zd 9771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐹‘𝑧) ∈ ℤ)
17	3	ffvelcdmda 5843	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐺‘𝑧) ∈ ℕ0)
18	17	nn0zd 9771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐺‘𝑧) ∈ ℤ)
19		zdceq 9725	. . . . . . . . . 10 ⊢ (((𝐹‘𝑧) ∈ ℤ ∧ (𝐺‘𝑧) ∈ ℤ) → DECID (𝐹‘𝑧) = (𝐺‘𝑧))
20	16, 18, 19	syl2anc 415	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → DECID (𝐹‘𝑧) = (𝐺‘𝑧))
21	12, 14, 20	ifcldcd 3678	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅) ∈ ω)
22	5, 9, 10, 21	fvmptd3 5799	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘𝑧) = if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅))
23		1lt2o 6715	. . . . . . . . . . . . 13 ⊢ 1o ∈ 2o
24	23	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → 1o ∈ 2o)
25		0lt2o 6714	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 2o
26	25	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → ∅ ∈ 2o)
27	1	ffvelcdmda 5843	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐹‘𝑥) ∈ ℕ0)
28	27	nn0zd 9771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐹‘𝑥) ∈ ℤ)
29	3	ffvelcdmda 5843	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐺‘𝑥) ∈ ℕ0)
30	29	nn0zd 9771	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐺‘𝑥) ∈ ℤ)
31		zdceq 9725	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ∈ ℤ ∧ (𝐺‘𝑥) ∈ ℤ) → DECID (𝐹‘𝑥) = (𝐺‘𝑥))
32	28, 30, 31	syl2anc 415	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → DECID (𝐹‘𝑥) = (𝐺‘𝑥))
33	24, 26, 32	ifcldcd 3678	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅) ∈ 2o)
34	33	fmpttd 5863	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅)):ℕ∞⟶2o)
35		2onn 6794	. . . . . . . . . . . 12 ⊢ 2o ∈ ω
36	35	elexi 2834	. . . . . . . . . . 11 ⊢ 2o ∈ V
37		nninfex 7462	. . . . . . . . . . 11 ⊢ ℕ∞ ∈ V
38	36, 37	elmap 6958	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅)) ∈ (2o ↑𝑚 ℕ∞) ↔ (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅)):ℕ∞⟶2o)
39	34, 38	sylibr 134	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅)) ∈ (2o ↑𝑚 ℕ∞))
40		fveq2 5695	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → (𝐹‘𝑥) = (𝐹‘(𝑤 ∈ ω ↦ 1o)))
41		fveq2 5695	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → (𝐺‘𝑥) = (𝐺‘(𝑤 ∈ ω ↦ 1o)))
42	40, 41	eqeq12d 2253	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o))))
43	42	ifbid 3662	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅) = if((𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)), 1o, ∅))
44		infnninf 7465	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ω ↦ 1o) ∈ ℕ∞
45	44	a1i 9	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ ω ↦ 1o) ∈ ℕ∞)
46		nninffeq.oo	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐹‘(𝑥 ∈ ω ↦ 1o)) = (𝐺‘(𝑥 ∈ ω ↦ 1o)))
47		eqidd 2239	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → 1o = 1o)
48	47	cbvmptv 4227	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω ↦ 1o) = (𝑤 ∈ ω ↦ 1o)
49	48	fveq2i 5698	. . . . . . . . . . . . . 14 ⊢ (𝐹‘(𝑥 ∈ ω ↦ 1o)) = (𝐹‘(𝑤 ∈ ω ↦ 1o))
50	48	fveq2i 5698	. . . . . . . . . . . . . 14 ⊢ (𝐺‘(𝑥 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o))
51	46, 49, 50	3eqtr3g 2294	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)))
52	51	iftrued 3647	. . . . . . . . . . . 12 ⊢ (𝜑 → if((𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)), 1o, ∅) = 1o)
53	52, 11	eqeltrdi 2329	. . . . . . . . . . 11 ⊢ (𝜑 → if((𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)), 1o, ∅) ∈ ω)
54	5, 43, 45, 53	fvmptd3 5799	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘(𝑤 ∈ ω ↦ 1o)) = if((𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)), 1o, ∅))
55	54, 52	eqtrd 2271	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘(𝑤 ∈ ω ↦ 1o)) = 1o)
56		nninffeq.n	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑛 ∈ ω (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
57		fveq2 5695	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) → (𝐹‘𝑥) = (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
58		fveq2 5695	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) → (𝐺‘𝑥) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
59	57, 58	eqeq12d 2253	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))))
60	59	ifbid 3662	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) → if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅) = if((𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))), 1o, ∅))
61		nnnninf 7467	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) ∈ ℕ∞)
62	61	ad2antlr 493	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) ∈ ℕ∞)
63		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) → (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
64	63	iftrued 3647	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) → if((𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))), 1o, ∅) = 1o)
65	64, 11	eqeltrdi 2329	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) → if((𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))), 1o, ∅) ∈ ω)
66	5, 60, 62, 65	fvmptd3 5799	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = if((𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))), 1o, ∅))
67	66, 64	eqtrd 2271	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ω) ∧ (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)))) → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)
68	67	ex 115	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ ω) → ((𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o))
69	68	ralimdva 2617	. . . . . . . . . 10 ⊢ (𝜑 → (∀𝑛 ∈ ω (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) → ∀𝑛 ∈ ω ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o))
70	56, 69	mpd 13	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ ω ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))) = 1o)
71	39, 55, 70	nninfall 17218	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ ℕ∞ ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘𝑧) = 1o)
72	71	r19.21bi 2638	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘𝑧) = 1o)
73	22, 72	eqtr3d 2273	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅) = 1o)
74	73	adantr 276	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅) = 1o)
75		simpr 110	. . . . . 6 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → ¬ (𝐹‘𝑧) = (𝐺‘𝑧))
76	75	iffalsed 3650	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅) = ∅)
77	74, 76	eqtr3d 2273	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → 1o = ∅)
78		1n0 6705	. . . . . 6 ⊢ 1o ≠ ∅
79	78	neii 2422	. . . . 5 ⊢ ¬ 1o = ∅
80	79	a1i 9	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → ¬ 1o = ∅)
81	77, 80	pm2.65da 671	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ¬ ¬ (𝐹‘𝑧) = (𝐺‘𝑧))
82		exmiddc 848	. . . 4 ⊢ (DECID (𝐹‘𝑧) = (𝐺‘𝑧) → ((𝐹‘𝑧) = (𝐺‘𝑧) ∨ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)))
83	20, 82	syl 14	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ((𝐹‘𝑧) = (𝐺‘𝑧) ∨ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)))
84	81, 83	ecased 1390	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐹‘𝑧) = (𝐺‘𝑧))
85	2, 4, 84	eqfnfvd 5809	1 ⊢ (𝜑 → 𝐹 = 𝐺)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 720  DECID wdc 846   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∅c0 3520  ifcif 3638   ↦ cmpt 4192  ωcom 4737  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085  1_oc1o 6680  2_oc2o 6681   ↑_𝑚 cmap 6922  ℕ_∞xnninf 7460  ℕ₀cn0 9568  ℤcz 9649

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-addcom 8280  ax-addass 8282  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-0id 8288  ax-rnegex 8289  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-ltadd 8296

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1o 6687  df-2o 6688  df-map 6924  df-nninf 7461  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-inn 9308  df-n0 9569  df-z 9650

This theorem is used by: (None)