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Theorem ennnfonelemg 13023

Description: Lemma for ennnfone 13045. 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 5643	. . . . . 6 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → (𝐹‘𝑦) = (𝐹‘𝑗))
5	3	imaeq2d 5076	. . . . . 6 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → (𝐹 “ 𝑦) = (𝐹 “ 𝑗))
6	4, 5	eleq12d 2302	. . . . 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 4933	. . . . . . . 8 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → dom 𝑥 = dom 𝑓)
9	8, 4	opeq12d 3870	. . . . . . 7 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → ⟨dom 𝑥, (𝐹‘𝑦)⟩ = ⟨dom 𝑓, (𝐹‘𝑗)⟩)
10	9	sneqd 3682	. . . . . 6 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → {⟨dom 𝑥, (𝐹‘𝑦)⟩} = {⟨dom 𝑓, (𝐹‘𝑗)⟩})
11	7, 10	uneq12d 3362	. . . . 5 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩}) = (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}))
12	6, 7, 11	ifbieq12d 3632	. . . 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 3312	. . . 4 ⊢ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ⊆ (𝐴 ↑_pm ω)
15		simprl 531	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
16	14, 15	sselid 3225	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑓 ∈ (𝐴 ↑_pm ω))
17		simprr 533	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑗 ∈ ω)
18		simplrl 537	. . . 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 4931	. . . . . 6 ⊢ (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) → dom 𝑔 = dom (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}))
20	19	eleq1d 2300	. . . . 5 ⊢ (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) → (dom 𝑔 ∈ ω ↔ dom (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) ∈ ω))
21		omex 4691	. . . . . . . 8 ⊢ ω ∈ V
22		ennnfonelemh.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
23		focdmex 6276	. . . . . . . 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 537	. . . . . . . 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 2959	. . . . . . . . . 10 ⊢ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} → 𝑓 ∈ (𝐴 ↑_pm ω))
29		elpmi 6835	. . . . . . . . . 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 4931	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
34	33	eleq1d 2300	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → (dom 𝑔 ∈ ω ↔ dom 𝑓 ∈ ω))
35	34	elrab 2962	. . . . . . . . 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 4710	. . . . . . . . 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 4640	. . . . . . . 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 5559	. . . . . . . . . 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 5783	. . . . . . . 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 5853	. . . . . . 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 1278	. . . . . 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 4468	. . . . . . . . 9 ⊢ suc dom 𝑓 = (dom 𝑓 ∪ {dom 𝑓})
50		peano2 4693	. . . . . . . . 9 ⊢ (dom 𝑓 ∈ ω → suc dom 𝑓 ∈ ω)
51	49, 50	eqeltrrid 2319	. . . . . . . 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 4703	. . . . . . 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 6834	. . . . . 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 1274	. . . . 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 5489	. . . . . 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 2308	. . . . 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 2964	. . . 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 13019	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → DECID (𝐹‘𝑗) ∈ (𝐹 “ 𝑗))
63	18, 59, 62	ifcldadc 3635	. . 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 6148	. 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 2308	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 841 = wceq 1397 ∈ wcel 2202 ≠ wne 2402 ∀wral 2510 ∃wrex 2511 {crab 2514 Vcvv 2802 ∪ cun 3198 ⊆ wss 3200 ∅c0 3494 ifcif 3605 {csn 3669 ⟨cop 3672 ↦ cmpt 4150 Ord word 4459 suc csuc 4462 ωcom 4688 ^◡ccnv 4724 dom cdm 4725 “ cima 4728 ⟶wf 5322 –onto→wfo 5324 ‘cfv 5326 (class class class)co 6017 ∈ cmpo 6019 freccfrec 6555 ↑_pm cpm 6817 0cc0 8031 1c1 8032 + caddc 8034 − cmin 8349 ℕ₀cn0 9401 ℤcz 9478 seqcseq 10708

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686

This theorem depends on definitions: df-bi 117 df-dc 842 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6020 df-oprab 6021 df-mpo 6022 df-pm 6819

This theorem is referenced by: ennnfonelemh 13024 ennnfonelem0 13025 ennnfonelemp1 13026 ennnfonelemom 13028

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