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Theorem ennnfonelemg 12889

Description: Lemma for ennnfone 12911. Closure for 𝐺. (Contributed by Jim Kingdon, 20-Jul-2023.)

**Hypotheses**
Ref	Expression
ennnfonelemh.dceq	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f	⊢ (𝜑 → 𝐹:ω–onto→𝐴)
ennnfonelemh.ne	⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
ennnfonelemh.g	⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑_pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
ennnfonelemh.n	⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
ennnfonelemh.j	⊢ 𝐽 = (𝑥 ∈ ℕ₀ ↦ if(𝑥 = 0, ∅, (^◡𝑁‘(𝑥 − 1))))
ennnfonelemh.h	⊢ 𝐻 = seq0(𝐺, 𝐽)

**Assertion**
Ref	Expression
ennnfonelemg	⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})

Distinct variable groups: 𝐴,𝑔,𝑥,𝑦 𝑔,𝐹,𝑥,𝑦 𝑥,𝑁 𝑓,𝑔,𝑥,𝑦 𝑔,𝑗,𝑥,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑓,𝑔,𝑗,𝑘,𝑛) 𝐴(𝑓,𝑗,𝑘,𝑛) 𝐹(𝑓,𝑗,𝑘,𝑛) 𝐺(𝑥,𝑦,𝑓,𝑔,𝑗,𝑘,𝑛) 𝐻(𝑥,𝑦,𝑓,𝑔,𝑗,𝑘,𝑛) 𝐽(𝑥,𝑦,𝑓,𝑔,𝑗,𝑘,𝑛) 𝑁(𝑦,𝑓,𝑔,𝑗,𝑘,𝑛)

**Proof of Theorem ennnfonelemg**
Step	Hyp	Ref	Expression
1		ennnfonelemh.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑_pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
2	1	a1i 9	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝐺 = (𝑥 ∈ (𝐴 ↑_pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩}))))
3		simpr 110	. . . . . . 7 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → 𝑦 = 𝑗)
4	3	fveq2d 5603	. . . . . 6 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → (𝐹‘𝑦) = (𝐹‘𝑗))
5	3	imaeq2d 5041	. . . . . 6 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → (𝐹 “ 𝑦) = (𝐹 “ 𝑗))
6	4, 5	eleq12d 2278	. . . . 5 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → ((𝐹‘𝑦) ∈ (𝐹 “ 𝑦) ↔ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)))
7		simpl 109	. . . . 5 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → 𝑥 = 𝑓)
8	7	dmeqd 4899	. . . . . . . 8 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → dom 𝑥 = dom 𝑓)
9	8, 4	opeq12d 3841	. . . . . . 7 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → ⟨dom 𝑥, (𝐹‘𝑦)⟩ = ⟨dom 𝑓, (𝐹‘𝑗)⟩)
10	9	sneqd 3656	. . . . . 6 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → {⟨dom 𝑥, (𝐹‘𝑦)⟩} = {⟨dom 𝑓, (𝐹‘𝑗)⟩})
11	7, 10	uneq12d 3336	. . . . 5 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩}) = (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}))
12	6, 7, 11	ifbieq12d 3606	. . . 4 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})) = if((𝐹‘𝑗) ∈ (𝐹 “ 𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩})))
13	12	adantl 277	. . 3 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ (𝑥 = 𝑓 ∧ 𝑦 = 𝑗)) → if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})) = if((𝐹‘𝑗) ∈ (𝐹 “ 𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩})))
14		ssrab2 3286	. . . 4 ⊢ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ⊆ (𝐴 ↑_pm ω)
15		simprl 529	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
16	14, 15	sselid 3199	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑓 ∈ (𝐴 ↑_pm ω))
17		simprr 531	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑗 ∈ ω)
18		simplrl 535	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → 𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
19		dmeq 4897	. . . . . 6 ⊢ (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) → dom 𝑔 = dom (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}))
20	19	eleq1d 2276	. . . . 5 ⊢ (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) → (dom 𝑔 ∈ ω ↔ dom (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) ∈ ω))
21		omex 4659	. . . . . . . 8 ⊢ ω ∈ V
22		ennnfonelemh.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
23		focdmex 6223	. . . . . . . 8 ⊢ (ω ∈ V → (𝐹:ω–onto→𝐴 → 𝐴 ∈ V))
24	21, 22, 23	mpsyl 65	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ V)
25	24	ad2antrr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → 𝐴 ∈ V)
26	21	a1i 9	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → ω ∈ V)
27		simplrl 535	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → 𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
28		elrabi 2933	. . . . . . . . . 10 ⊢ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} → 𝑓 ∈ (𝐴 ↑_pm ω))
29		elpmi 6777	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐴 ↑_pm ω) → (𝑓:dom 𝑓⟶𝐴 ∧ dom 𝑓 ⊆ ω))
30	28, 29	syl 14	. . . . . . . . 9 ⊢ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} → (𝑓:dom 𝑓⟶𝐴 ∧ dom 𝑓 ⊆ ω))
31	30	simpld 112	. . . . . . . 8 ⊢ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} → 𝑓:dom 𝑓⟶𝐴)
32	27, 31	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → 𝑓:dom 𝑓⟶𝐴)
33		dmeq 4897	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
34	33	eleq1d 2276	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → (dom 𝑔 ∈ ω ↔ dom 𝑓 ∈ ω))
35	34	elrab 2936	. . . . . . . . 9 ⊢ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ↔ (𝑓 ∈ (𝐴 ↑_pm ω) ∧ dom 𝑓 ∈ ω))
36	35	simprbi 275	. . . . . . . 8 ⊢ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} → dom 𝑓 ∈ ω)
37	27, 36	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → dom 𝑓 ∈ ω)
38		nnord 4678	. . . . . . . . 9 ⊢ (dom 𝑓 ∈ ω → Ord dom 𝑓)
39	37, 38	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → Ord dom 𝑓)
40		ordirr 4608	. . . . . . . 8 ⊢ (Ord dom 𝑓 → ¬ dom 𝑓 ∈ dom 𝑓)
41	39, 40	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → ¬ dom 𝑓 ∈ dom 𝑓)
42	22	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝐹:ω–onto→𝐴)
43		fof 5520	. . . . . . . . . 10 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴)
44	42, 43	syl 14	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝐹:ω⟶𝐴)
45	44, 17	ffvelcdmd 5739	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝐹‘𝑗) ∈ 𝐴)
46	45	adantr 276	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → (𝐹‘𝑗) ∈ 𝐴)
47		fsnunf 5807	. . . . . . 7 ⊢ ((𝑓:dom 𝑓⟶𝐴 ∧ (dom 𝑓 ∈ ω ∧ ¬ dom 𝑓 ∈ dom 𝑓) ∧ (𝐹‘𝑗) ∈ 𝐴) → (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}):(dom 𝑓 ∪ {dom 𝑓})⟶𝐴)
48	32, 37, 41, 46, 47	syl121anc 1255	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}):(dom 𝑓 ∪ {dom 𝑓})⟶𝐴)
49		df-suc 4436	. . . . . . . . 9 ⊢ suc dom 𝑓 = (dom 𝑓 ∪ {dom 𝑓})
50		peano2 4661	. . . . . . . . 9 ⊢ (dom 𝑓 ∈ ω → suc dom 𝑓 ∈ ω)
51	49, 50	eqeltrrid 2295	. . . . . . . 8 ⊢ (dom 𝑓 ∈ ω → (dom 𝑓 ∪ {dom 𝑓}) ∈ ω)
52	37, 51	syl 14	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → (dom 𝑓 ∪ {dom 𝑓}) ∈ ω)
53		elomssom 4671	. . . . . . 7 ⊢ ((dom 𝑓 ∪ {dom 𝑓}) ∈ ω → (dom 𝑓 ∪ {dom 𝑓}) ⊆ ω)
54	52, 53	syl 14	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → (dom 𝑓 ∪ {dom 𝑓}) ⊆ ω)
55		elpm2r 6776	. . . . . 6 ⊢ (((𝐴 ∈ V ∧ ω ∈ V) ∧ ((𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}):(dom 𝑓 ∪ {dom 𝑓})⟶𝐴 ∧ (dom 𝑓 ∪ {dom 𝑓}) ⊆ ω)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) ∈ (𝐴 ↑_pm ω))
56	25, 26, 48, 54, 55	syl22anc 1251	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) ∈ (𝐴 ↑_pm ω))
57	48	fdmd 5452	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → dom (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) = (dom 𝑓 ∪ {dom 𝑓}))
58	57, 52	eqeltrd 2284	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → dom (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) ∈ ω)
59	20, 56, 58	elrabd 2938	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹‘𝑗) ∈ (𝐹 “ 𝑗)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
60		ennnfonelemh.dceq	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
61	60	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
62	61, 42, 17	ennnfonelemdc 12885	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → DECID (𝐹‘𝑗) ∈ (𝐹 “ 𝑗))
63	18, 59, 62	ifcldadc 3609	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → if((𝐹‘𝑗) ∈ (𝐹 “ 𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩})) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
64	2, 13, 16, 17, 63	ovmpod 6096	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) = if((𝐹‘𝑗) ∈ (𝐹 “ 𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩})))
65	64, 63	eqeltrd 2284	1 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 DECID wdc 836 = wceq 1373 ∈ wcel 2178 ≠ wne 2378 ∀wral 2486 ∃wrex 2487 {crab 2490 Vcvv 2776 ∪ cun 3172 ⊆ wss 3174 ∅c0 3468 ifcif 3579 {csn 3643 ⟨cop 3646 ↦ cmpt 4121 Ord word 4427 suc csuc 4430 ωcom 4656 ^◡ccnv 4692 dom cdm 4693 “ cima 4696 ⟶wf 5286 –onto→wfo 5288 ‘cfv 5290 (class class class)co 5967 ∈ cmpo 5969 freccfrec 6499 ↑_pm cpm 6759 0cc0 7960 1c1 7961 + caddc 7963 − cmin 8278 ℕ₀cn0 9330 ℤcz 9407 seqcseq 10629

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654

This theorem depends on definitions: df-bi 117 df-dc 837 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-iord 4431 df-on 4433 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-ov 5970 df-oprab 5971 df-mpo 5972 df-pm 6761

This theorem is referenced by: ennnfonelemh 12890 ennnfonelem0 12891 ennnfonelemp1 12892 ennnfonelemom 12894

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