Theorem nninffeq 15580

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 5405	. 2 ⊢ (𝜑 → 𝐹 Fn ℕ∞)
3		nninffeq.g	. . 3 ⊢ (𝜑 → 𝐺:ℕ∞⟶ℕ0)
4	3	ffnd 5405	. 2 ⊢ (𝜑 → 𝐺 Fn ℕ∞)
5		eqid 2193	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅)) = (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))
6		fveq2 5555	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
7		fveq2 5555	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
8	6, 7	eqeq12d 2208	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝑧) = (𝐺‘𝑧)))
9	8	ifbid 3579	. . . . . . . 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 6575	. . . . . . . . . 10 ⊢ 1o ∈ ω
12	11	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → 1o ∈ ω)
13		peano1 4627	. . . . . . . . . 10 ⊢ ∅ ∈ ω
14	13	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ∅ ∈ ω)
15	1	ffvelcdmda 5694	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐹‘𝑧) ∈ ℕ0)
16	15	nn0zd 9440	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐹‘𝑧) ∈ ℤ)
17	3	ffvelcdmda 5694	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐺‘𝑧) ∈ ℕ0)
18	17	nn0zd 9440	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐺‘𝑧) ∈ ℤ)
19		zdceq 9395	. . . . . . . . . 10 ⊢ (((𝐹‘𝑧) ∈ ℤ ∧ (𝐺‘𝑧) ∈ ℤ) → DECID (𝐹‘𝑧) = (𝐺‘𝑧))
20	16, 18, 19	syl2anc 411	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → DECID (𝐹‘𝑧) = (𝐺‘𝑧))
21	12, 14, 20	ifcldcd 3594	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅) ∈ ω)
22	5, 9, 10, 21	fvmptd3 5652	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘𝑧) = if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅))
23		1lt2o 6497	. . . . . . . . . . . . 13 ⊢ 1o ∈ 2o
24	23	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → 1o ∈ 2o)
25		0lt2o 6496	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 2o
26	25	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → ∅ ∈ 2o)
27	1	ffvelcdmda 5694	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐹‘𝑥) ∈ ℕ0)
28	27	nn0zd 9440	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐹‘𝑥) ∈ ℤ)
29	3	ffvelcdmda 5694	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐺‘𝑥) ∈ ℕ0)
30	29	nn0zd 9440	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐺‘𝑥) ∈ ℤ)
31		zdceq 9395	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥) ∈ ℤ ∧ (𝐺‘𝑥) ∈ ℤ) → DECID (𝐹‘𝑥) = (𝐺‘𝑥))
32	28, 30, 31	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → DECID (𝐹‘𝑥) = (𝐺‘𝑥))
33	24, 26, 32	ifcldcd 3594	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅) ∈ 2o)
34	33	fmpttd 5714	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅)):ℕ∞⟶2o)
35		2onn 6576	. . . . . . . . . . . 12 ⊢ 2o ∈ ω
36	35	elexi 2772	. . . . . . . . . . 11 ⊢ 2o ∈ V
37		nninfex 7182	. . . . . . . . . . 11 ⊢ ℕ∞ ∈ V
38	36, 37	elmap 6733	. . . . . . . . . 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 5555	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → (𝐹‘𝑥) = (𝐹‘(𝑤 ∈ ω ↦ 1o)))
41		fveq2 5555	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → (𝐺‘𝑥) = (𝐺‘(𝑤 ∈ ω ↦ 1o)))
42	40, 41	eqeq12d 2208	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o))))
43	42	ifbid 3579	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅) = if((𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)), 1o, ∅))
44		infnninf 7185	. . . . . . . . . . . 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 2194	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → 1o = 1o)
48	47	cbvmptv 4126	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω ↦ 1o) = (𝑤 ∈ ω ↦ 1o)
49	48	fveq2i 5558	. . . . . . . . . . . . . 14 ⊢ (𝐹‘(𝑥 ∈ ω ↦ 1o)) = (𝐹‘(𝑤 ∈ ω ↦ 1o))
50	48	fveq2i 5558	. . . . . . . . . . . . . 14 ⊢ (𝐺‘(𝑥 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o))
51	46, 49, 50	3eqtr3g 2249	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)))
52	51	iftrued 3565	. . . . . . . . . . . 12 ⊢ (𝜑 → if((𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)), 1o, ∅) = 1o)
53	52, 11	eqeltrdi 2284	. . . . . . . . . . 11 ⊢ (𝜑 → if((𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)), 1o, ∅) ∈ ω)
54	5, 43, 45, 53	fvmptd3 5652	. . . . . . . . . 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 2226	. . . . . . . . 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 5555	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) → (𝐹‘𝑥) = (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
58		fveq2 5555	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) → (𝐺‘𝑥) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
59	57, 58	eqeq12d 2208	. . . . . . . . . . . . . . 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 3579	. . . . . . . . . . . . . 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 7187	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ω → (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) ∈ ℕ∞)
62	61	ad2antlr 489	. . . . . . . . . . . . . 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 3565	. . . . . . . . . . . . . . 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 2284	. . . . . . . . . . . . . 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 5652	. . . . . . . . . . . . 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 2226	. . . . . . . . . . . 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 2561	. . . . . . . . . 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 15569	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ ℕ∞ ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘𝑧) = 1o)
72	71	r19.21bi 2582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘𝑧) = 1o)
73	22, 72	eqtr3d 2228	. . . . . 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 3568	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅) = ∅)
77	74, 76	eqtr3d 2228	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → 1o = ∅)
78		1n0 6487	. . . . . 6 ⊢ 1o ≠ ∅
79	78	neii 2366	. . . . 5 ⊢ ¬ 1o = ∅
80	79	a1i 9	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → ¬ 1o = ∅)
81	77, 80	pm2.65da 662	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ¬ ¬ (𝐹‘𝑧) = (𝐺‘𝑧))
82		exmiddc 837	. . . 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 1360	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐹‘𝑧) = (𝐺‘𝑧))
85	2, 4, 84	eqfnfvd 5659	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∅c0 3447  ifcif 3558   ↦ cmpt 4091  ωcom 4623  ⟶wf 5251  ‘cfv 5255  (class class class)co 5919  1_oc1o 6464  2_oc2o 6465   ↑_𝑚 cmap 6704  ℕ_∞xnninf 7180  ℕ₀cn0 9243  ℤcz 9320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-ltadd 7990

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1o 6471  df-2o 6472  df-map 6706  df-nninf 7181  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-inn 8985  df-n0 9244  df-z 9321

This theorem is referenced by: (None)