Theorem nninffeq 14739

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 5366	. 2 ⊢ (𝜑 → 𝐹 Fn ℕ∞)
3		nninffeq.g	. . 3 ⊢ (𝜑 → 𝐺:ℕ∞⟶ℕ0)
4	3	ffnd 5366	. 2 ⊢ (𝜑 → 𝐺 Fn ℕ∞)
5		eqid 2177	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅)) = (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))
6		fveq2 5515	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
7		fveq2 5515	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝐺‘𝑥) = (𝐺‘𝑧))
8	6, 7	eqeq12d 2192	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝑧) = (𝐺‘𝑧)))
9	8	ifbid 3555	. . . . . . . 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 6520	. . . . . . . . . 10 ⊢ 1o ∈ ω
12	11	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → 1o ∈ ω)
13		peano1 4593	. . . . . . . . . 10 ⊢ ∅ ∈ ω
14	13	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ∅ ∈ ω)
15	1	ffvelcdmda 5651	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐹‘𝑧) ∈ ℕ0)
16	15	nn0zd 9372	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐹‘𝑧) ∈ ℤ)
17	3	ffvelcdmda 5651	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐺‘𝑧) ∈ ℕ0)
18	17	nn0zd 9372	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐺‘𝑧) ∈ ℤ)
19		zdceq 9327	. . . . . . . . . 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 3570	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅) ∈ ω)
22	5, 9, 10, 21	fvmptd3 5609	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘𝑧) = if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅))
23		1lt2o 6442	. . . . . . . . . . . . 13 ⊢ 1o ∈ 2o
24	23	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → 1o ∈ 2o)
25		0lt2o 6441	. . . . . . . . . . . . 13 ⊢ ∅ ∈ 2o
26	25	a1i 9	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → ∅ ∈ 2o)
27	1	ffvelcdmda 5651	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐹‘𝑥) ∈ ℕ0)
28	27	nn0zd 9372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐹‘𝑥) ∈ ℤ)
29	3	ffvelcdmda 5651	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐺‘𝑥) ∈ ℕ0)
30	29	nn0zd 9372	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → (𝐺‘𝑥) ∈ ℤ)
31		zdceq 9327	. . . . . . . . . . . . 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 3570	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℕ∞) → if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅) ∈ 2o)
34	33	fmpttd 5671	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅)):ℕ∞⟶2o)
35		2onn 6521	. . . . . . . . . . . 12 ⊢ 2o ∈ ω
36	35	elexi 2749	. . . . . . . . . . 11 ⊢ 2o ∈ V
37		nninfex 7119	. . . . . . . . . . 11 ⊢ ℕ∞ ∈ V
38	36, 37	elmap 6676	. . . . . . . . . 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 5515	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → (𝐹‘𝑥) = (𝐹‘(𝑤 ∈ ω ↦ 1o)))
41		fveq2 5515	. . . . . . . . . . . . 13 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → (𝐺‘𝑥) = (𝐺‘(𝑤 ∈ ω ↦ 1o)))
42	40, 41	eqeq12d 2192	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o))))
43	42	ifbid 3555	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑤 ∈ ω ↦ 1o) → if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅) = if((𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)), 1o, ∅))
44		infnninf 7121	. . . . . . . . . . . 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 2178	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑤 → 1o = 1o)
48	47	cbvmptv 4099	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω ↦ 1o) = (𝑤 ∈ ω ↦ 1o)
49	48	fveq2i 5518	. . . . . . . . . . . . . 14 ⊢ (𝐹‘(𝑥 ∈ ω ↦ 1o)) = (𝐹‘(𝑤 ∈ ω ↦ 1o))
50	48	fveq2i 5518	. . . . . . . . . . . . . 14 ⊢ (𝐺‘(𝑥 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o))
51	46, 49, 50	3eqtr3g 2233	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)))
52	51	iftrued 3541	. . . . . . . . . . . 12 ⊢ (𝜑 → if((𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)), 1o, ∅) = 1o)
53	52, 11	eqeltrdi 2268	. . . . . . . . . . 11 ⊢ (𝜑 → if((𝐹‘(𝑤 ∈ ω ↦ 1o)) = (𝐺‘(𝑤 ∈ ω ↦ 1o)), 1o, ∅) ∈ ω)
54	5, 43, 45, 53	fvmptd3 5609	. . . . . . . . . 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 2210	. . . . . . . . 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 5515	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) → (𝐹‘𝑥) = (𝐹‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
58		fveq2 5515	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅)) → (𝐺‘𝑥) = (𝐺‘(𝑖 ∈ ω ↦ if(𝑖 ∈ 𝑛, 1o, ∅))))
59	57, 58	eqeq12d 2192	. . . . . . . . . . . . . . 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 3555	. . . . . . . . . . . . . 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 7123	. . . . . . . . . . . . . . 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 3541	. . . . . . . . . . . . . . 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 2268	. . . . . . . . . . . . . 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 5609	. . . . . . . . . . . . 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 2210	. . . . . . . . . . . 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 2544	. . . . . . . . . 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 14728	. . . . . . . 8 ⊢ (𝜑 → ∀𝑧 ∈ ℕ∞ ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘𝑧) = 1o)
72	71	r19.21bi 2565	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ((𝑥 ∈ ℕ∞ ↦ if((𝐹‘𝑥) = (𝐺‘𝑥), 1o, ∅))‘𝑧) = 1o)
73	22, 72	eqtr3d 2212	. . . . . 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 3544	. . . . 5 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → if((𝐹‘𝑧) = (𝐺‘𝑧), 1o, ∅) = ∅)
77	74, 76	eqtr3d 2212	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → 1o = ∅)
78		1n0 6432	. . . . . 6 ⊢ 1o ≠ ∅
79	78	neii 2349	. . . . 5 ⊢ ¬ 1o = ∅
80	79	a1i 9	. . . 4 ⊢ (((𝜑 ∧ 𝑧 ∈ ℕ∞) ∧ ¬ (𝐹‘𝑧) = (𝐺‘𝑧)) → ¬ 1o = ∅)
81	77, 80	pm2.65da 661	. . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → ¬ ¬ (𝐹‘𝑧) = (𝐺‘𝑧))
82		exmiddc 836	. . . 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 1349	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ℕ∞) → (𝐹‘𝑧) = (𝐺‘𝑧))
85	2, 4, 84	eqfnfvd 5616	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∅c0 3422  ifcif 3534   ↦ cmpt 4064  ωcom 4589  ⟶wf 5212  ‘cfv 5216  (class class class)co 5874  1_oc1o 6409  2_oc2o 6410   ↑_𝑚 cmap 6647  ℕ_∞xnninf 7117  ℕ₀cn0 9175  ℤcz 9252

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1o 6416  df-2o 6417  df-map 6649  df-nninf 7118  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253

This theorem is referenced by: (None)