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Theorem ennnfonelemg 12645

Description: Lemma for ennnfone 12667. 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 5565	. . . . . 6 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → (𝐹‘𝑦) = (𝐹‘𝑗))
5	3	imaeq2d 5010	. . . . . 6 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → (𝐹 “ 𝑦) = (𝐹 “ 𝑗))
6	4, 5	eleq12d 2267	. . . . 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 4869	. . . . . . . 8 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → dom 𝑥 = dom 𝑓)
9	8, 4	opeq12d 3817	. . . . . . 7 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → ⟨dom 𝑥, (𝐹‘𝑦)⟩ = ⟨dom 𝑓, (𝐹‘𝑗)⟩)
10	9	sneqd 3636	. . . . . 6 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → {⟨dom 𝑥, (𝐹‘𝑦)⟩} = {⟨dom 𝑓, (𝐹‘𝑗)⟩})
11	7, 10	uneq12d 3319	. . . . 5 ⊢ ((𝑥 = 𝑓 ∧ 𝑦 = 𝑗) → (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩}) = (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}))
12	6, 7, 11	ifbieq12d 3588	. . . 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 3269	. . . 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 3182	. . 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 4867	. . . . . 6 ⊢ (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) → dom 𝑔 = dom (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}))
20	19	eleq1d 2265	. . . . 5 ⊢ (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) → (dom 𝑔 ∈ ω ↔ dom (𝑓 ∪ {⟨dom 𝑓, (𝐹‘𝑗)⟩}) ∈ ω))
21		omex 4630	. . . . . . . 8 ⊢ ω ∈ V
22		ennnfonelemh.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
23		focdmex 6181	. . . . . . . 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 2917	. . . . . . . . . 10 ⊢ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} → 𝑓 ∈ (𝐴 ↑_pm ω))
29		elpmi 6735	. . . . . . . . . 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 4867	. . . . . . . . . . 11 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
34	33	eleq1d 2265	. . . . . . . . . 10 ⊢ (𝑔 = 𝑓 → (dom 𝑔 ∈ ω ↔ dom 𝑓 ∈ ω))
35	34	elrab 2920	. . . . . . . . 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 4649	. . . . . . . . 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 4579	. . . . . . . 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 5483	. . . . . . . . . 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 5701	. . . . . . . 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 5765	. . . . . . 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 1254	. . . . . 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 4407	. . . . . . . . 9 ⊢ suc dom 𝑓 = (dom 𝑓 ∪ {dom 𝑓})
50		peano2 4632	. . . . . . . . 9 ⊢ (dom 𝑓 ∈ ω → suc dom 𝑓 ∈ ω)
51	49, 50	eqeltrrid 2284	. . . . . . . 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 4642	. . . . . . 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 6734	. . . . . 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 1250	. . . . 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 5417	. . . . . 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 2273	. . . . 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 2922	. . . 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 12641	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → DECID (𝐹‘𝑗) ∈ (𝐹 “ 𝑗))
63	18, 59, 62	ifcldadc 3591	. . 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 6054	. 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 2273	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 835 = wceq 1364 ∈ wcel 2167 ≠ wne 2367 ∀wral 2475 ∃wrex 2476 {crab 2479 Vcvv 2763 ∪ cun 3155 ⊆ wss 3157 ∅c0 3451 ifcif 3562 {csn 3623 ⟨cop 3626 ↦ cmpt 4095 Ord word 4398 suc csuc 4401 ωcom 4627 ^◡ccnv 4663 dom cdm 4664 “ cima 4667 ⟶wf 5255 –onto→wfo 5257 ‘cfv 5259 (class class class)co 5925 ∈ cmpo 5927 freccfrec 6457 ↑_pm cpm 6717 0cc0 7896 1c1 7897 + caddc 7899 − cmin 8214 ℕ₀cn0 9266 ℤcz 9343 seqcseq 10556

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625

This theorem depends on definitions: df-bi 117 df-dc 836 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-iord 4402 df-on 4404 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pm 6719

This theorem is referenced by: ennnfonelemh 12646 ennnfonelem0 12647 ennnfonelemp1 12648 ennnfonelemom 12650

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