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Theorem ennnfonelemg 11761
Description: Lemma for ennnfone 11783. Closure for 𝐺. (Contributed by Jim Kingdon, 20-Jul-2023.)
Hypotheses
Ref Expression
ennnfonelemh.dceq (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f (𝜑𝐹:ω–onto𝐴)
ennnfonelemh.ne (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
ennnfonelemh.g 𝐺 = (𝑥 ∈ (𝐴pm ω), 𝑦 ∈ ω ↦ if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})))
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 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴pm 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
StepHypRef Expression
1 ennnfonelemh.g . . . 4 𝐺 = (𝑥 ∈ (𝐴pm ω), 𝑦 ∈ ω ↦ if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})))
21a1i 9 . . 3 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝐺 = (𝑥 ∈ (𝐴pm ω), 𝑦 ∈ ω ↦ if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩}))))
3 simpr 109 . . . . . . 7 ((𝑥 = 𝑓𝑦 = 𝑗) → 𝑦 = 𝑗)
43fveq2d 5379 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝐹𝑦) = (𝐹𝑗))
53imaeq2d 4839 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝐹𝑦) = (𝐹𝑗))
64, 5eleq12d 2185 . . . . 5 ((𝑥 = 𝑓𝑦 = 𝑗) → ((𝐹𝑦) ∈ (𝐹𝑦) ↔ (𝐹𝑗) ∈ (𝐹𝑗)))
7 simpl 108 . . . . 5 ((𝑥 = 𝑓𝑦 = 𝑗) → 𝑥 = 𝑓)
87dmeqd 4701 . . . . . . . 8 ((𝑥 = 𝑓𝑦 = 𝑗) → dom 𝑥 = dom 𝑓)
98, 4opeq12d 3679 . . . . . . 7 ((𝑥 = 𝑓𝑦 = 𝑗) → ⟨dom 𝑥, (𝐹𝑦)⟩ = ⟨dom 𝑓, (𝐹𝑗)⟩)
109sneqd 3506 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → {⟨dom 𝑥, (𝐹𝑦)⟩} = {⟨dom 𝑓, (𝐹𝑗)⟩})
117, 10uneq12d 3197 . . . . 5 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩}) = (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}))
126, 7, 11ifbieq12d 3464 . . . 4 ((𝑥 = 𝑓𝑦 = 𝑗) → if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})) = if((𝐹𝑗) ∈ (𝐹𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩})))
1312adantl 273 . . 3 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ (𝑥 = 𝑓𝑦 = 𝑗)) → if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})) = if((𝐹𝑗) ∈ (𝐹𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩})))
14 ssrab2 3148 . . . 4 {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ⊆ (𝐴pm ω)
15 simprl 503 . . . 4 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω})
1614, 15sseldi 3061 . . 3 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑓 ∈ (𝐴pm ω))
17 simprr 504 . . 3 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑗 ∈ ω)
18 simplrl 507 . . . 4 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ (𝐹𝑗) ∈ (𝐹𝑗)) → 𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω})
19 dmeq 4699 . . . . . 6 (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) → dom 𝑔 = dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}))
2019eleq1d 2183 . . . . 5 (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) → (dom 𝑔 ∈ ω ↔ dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ ω))
21 omex 4467 . . . . . . . 8 ω ∈ V
22 ennnfonelemh.f . . . . . . . 8 (𝜑𝐹:ω–onto𝐴)
23 focdmex 10426 . . . . . . . 8 ((ω ∈ V ∧ 𝐹:ω–onto𝐴) → 𝐴 ∈ V)
2421, 22, 23sylancr 408 . . . . . . 7 (𝜑𝐴 ∈ V)
2524ad2antrr 477 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → 𝐴 ∈ V)
2621a1i 9 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → ω ∈ V)
27 simplrl 507 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → 𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω})
28 elrabi 2806 . . . . . . . . . 10 (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} → 𝑓 ∈ (𝐴pm ω))
29 elpmi 6515 . . . . . . . . . 10 (𝑓 ∈ (𝐴pm ω) → (𝑓:dom 𝑓𝐴 ∧ dom 𝑓 ⊆ ω))
3028, 29syl 14 . . . . . . . . 9 (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} → (𝑓:dom 𝑓𝐴 ∧ dom 𝑓 ⊆ ω))
3130simpld 111 . . . . . . . 8 (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} → 𝑓:dom 𝑓𝐴)
3227, 31syl 14 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → 𝑓:dom 𝑓𝐴)
33 dmeq 4699 . . . . . . . . . . 11 (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
3433eleq1d 2183 . . . . . . . . . 10 (𝑔 = 𝑓 → (dom 𝑔 ∈ ω ↔ dom 𝑓 ∈ ω))
3534elrab 2809 . . . . . . . . 9 (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ↔ (𝑓 ∈ (𝐴pm ω) ∧ dom 𝑓 ∈ ω))
3635simprbi 271 . . . . . . . 8 (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} → dom 𝑓 ∈ ω)
3727, 36syl 14 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → dom 𝑓 ∈ ω)
38 nnord 4485 . . . . . . . . 9 (dom 𝑓 ∈ ω → Ord dom 𝑓)
3937, 38syl 14 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → Ord dom 𝑓)
40 ordirr 4417 . . . . . . . 8 (Ord dom 𝑓 → ¬ dom 𝑓 ∈ dom 𝑓)
4139, 40syl 14 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → ¬ dom 𝑓 ∈ dom 𝑓)
4222adantr 272 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝐹:ω–onto𝐴)
43 fof 5303 . . . . . . . . . 10 (𝐹:ω–onto𝐴𝐹:ω⟶𝐴)
4442, 43syl 14 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝐹:ω⟶𝐴)
4544, 17ffvelrnd 5510 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝐹𝑗) ∈ 𝐴)
4645adantr 272 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (𝐹𝑗) ∈ 𝐴)
47 fsnunf 5574 . . . . . . 7 ((𝑓:dom 𝑓𝐴 ∧ (dom 𝑓 ∈ ω ∧ ¬ dom 𝑓 ∈ dom 𝑓) ∧ (𝐹𝑗) ∈ 𝐴) → (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}):(dom 𝑓 ∪ {dom 𝑓})⟶𝐴)
4832, 37, 41, 46, 47syl121anc 1204 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}):(dom 𝑓 ∪ {dom 𝑓})⟶𝐴)
49 df-suc 4253 . . . . . . . . 9 suc dom 𝑓 = (dom 𝑓 ∪ {dom 𝑓})
50 peano2 4469 . . . . . . . . 9 (dom 𝑓 ∈ ω → suc dom 𝑓 ∈ ω)
5149, 50syl5eqelr 2202 . . . . . . . 8 (dom 𝑓 ∈ ω → (dom 𝑓 ∪ {dom 𝑓}) ∈ ω)
5237, 51syl 14 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (dom 𝑓 ∪ {dom 𝑓}) ∈ ω)
53 omelon 4482 . . . . . . . 8 ω ∈ On
5453onelssi 4311 . . . . . . 7 ((dom 𝑓 ∪ {dom 𝑓}) ∈ ω → (dom 𝑓 ∪ {dom 𝑓}) ⊆ ω)
5552, 54syl 14 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (dom 𝑓 ∪ {dom 𝑓}) ⊆ ω)
56 elpm2r 6514 . . . . . 6 (((𝐴 ∈ V ∧ ω ∈ V) ∧ ((𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}):(dom 𝑓 ∪ {dom 𝑓})⟶𝐴 ∧ (dom 𝑓 ∪ {dom 𝑓}) ⊆ ω)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ (𝐴pm ω))
5725, 26, 48, 55, 56syl22anc 1200 . . . . 5 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ (𝐴pm ω))
5848fdmd 5237 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) = (dom 𝑓 ∪ {dom 𝑓}))
5958, 52eqeltrd 2191 . . . . 5 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ ω)
6020, 57, 59elrabd 2811 . . . 4 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω})
61 ennnfonelemh.dceq . . . . . 6 (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
6261adantr 272 . . . . 5 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
6362, 42, 17ennnfonelemdc 11757 . . . 4 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → DECID (𝐹𝑗) ∈ (𝐹𝑗))
6418, 60, 63ifcldadc 3467 . . 3 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → if((𝐹𝑗) ∈ (𝐹𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩})) ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω})
652, 13, 16, 17, 64ovmpod 5852 . 2 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) = if((𝐹𝑗) ∈ (𝐹𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩})))
6665, 64eqeltrd 2191 1 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  DECID wdc 802   = wceq 1314  wcel 1463  wne 2282  wral 2390  wrex 2391  {crab 2394  Vcvv 2657  cun 3035  wss 3037  c0 3329  ifcif 3440  {csn 3493  cop 3496  cmpt 3949  Ord word 4244  suc csuc 4247  ωcom 4464  ccnv 4498  dom cdm 4499  cima 4502  wf 5077  ontowfo 5079  cfv 5081  (class class class)co 5728  cmpo 5730  freccfrec 6241  pm cpm 6497  0cc0 7547  1c1 7548   + caddc 7550  cmin 7856  0cn0 8881  cz 8958  seqcseq 10111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-pm 6499
This theorem is referenced by:  ennnfonelemh  11762  ennnfonelem0  11763  ennnfonelemp1  11764  ennnfonelemom  11766
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