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Theorem ennnfonelemg 12404
Description: Lemma for ennnfone 12426. 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 110 . . . . . . 7 ((𝑥 = 𝑓𝑦 = 𝑗) → 𝑦 = 𝑗)
43fveq2d 5520 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝐹𝑦) = (𝐹𝑗))
53imaeq2d 4971 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝐹𝑦) = (𝐹𝑗))
64, 5eleq12d 2248 . . . . 5 ((𝑥 = 𝑓𝑦 = 𝑗) → ((𝐹𝑦) ∈ (𝐹𝑦) ↔ (𝐹𝑗) ∈ (𝐹𝑗)))
7 simpl 109 . . . . 5 ((𝑥 = 𝑓𝑦 = 𝑗) → 𝑥 = 𝑓)
87dmeqd 4830 . . . . . . . 8 ((𝑥 = 𝑓𝑦 = 𝑗) → dom 𝑥 = dom 𝑓)
98, 4opeq12d 3787 . . . . . . 7 ((𝑥 = 𝑓𝑦 = 𝑗) → ⟨dom 𝑥, (𝐹𝑦)⟩ = ⟨dom 𝑓, (𝐹𝑗)⟩)
109sneqd 3606 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → {⟨dom 𝑥, (𝐹𝑦)⟩} = {⟨dom 𝑓, (𝐹𝑗)⟩})
117, 10uneq12d 3291 . . . . 5 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩}) = (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}))
126, 7, 11ifbieq12d 3561 . . . 4 ((𝑥 = 𝑓𝑦 = 𝑗) → if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})) = if((𝐹𝑗) ∈ (𝐹𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩})))
1312adantl 277 . . 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 3241 . . . 4 {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ⊆ (𝐴pm ω)
15 simprl 529 . . . 4 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω})
1614, 15sselid 3154 . . 3 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑓 ∈ (𝐴pm ω))
17 simprr 531 . . 3 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝑗 ∈ ω)
18 simplrl 535 . . . 4 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ (𝐹𝑗) ∈ (𝐹𝑗)) → 𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω})
19 dmeq 4828 . . . . . 6 (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) → dom 𝑔 = dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}))
2019eleq1d 2246 . . . . 5 (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) → (dom 𝑔 ∈ ω ↔ dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ ω))
21 omex 4593 . . . . . . . 8 ω ∈ V
22 ennnfonelemh.f . . . . . . . 8 (𝜑𝐹:ω–onto𝐴)
23 focdmex 6116 . . . . . . . 8 (ω ∈ V → (𝐹:ω–onto𝐴𝐴 ∈ V))
2421, 22, 23mpsyl 65 . . . . . . 7 (𝜑𝐴 ∈ V)
2524ad2antrr 488 . . . . . 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 535 . . . . . . . 8 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → 𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω})
28 elrabi 2891 . . . . . . . . . 10 (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} → 𝑓 ∈ (𝐴pm ω))
29 elpmi 6667 . . . . . . . . . 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 112 . . . . . . . 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 4828 . . . . . . . . . . 11 (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
3433eleq1d 2246 . . . . . . . . . 10 (𝑔 = 𝑓 → (dom 𝑔 ∈ ω ↔ dom 𝑓 ∈ ω))
3534elrab 2894 . . . . . . . . 9 (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ↔ (𝑓 ∈ (𝐴pm ω) ∧ dom 𝑓 ∈ ω))
3635simprbi 275 . . . . . . . 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 4612 . . . . . . . . 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 4542 . . . . . . . 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 276 . . . . . . . . . 10 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝐹:ω–onto𝐴)
43 fof 5439 . . . . . . . . . 10 (𝐹:ω–onto𝐴𝐹:ω⟶𝐴)
4442, 43syl 14 . . . . . . . . 9 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → 𝐹:ω⟶𝐴)
4544, 17ffvelcdmd 5653 . . . . . . . 8 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝐹𝑗) ∈ 𝐴)
4645adantr 276 . . . . . . 7 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (𝐹𝑗) ∈ 𝐴)
47 fsnunf 5717 . . . . . . 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 1243 . . . . . 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 4372 . . . . . . . . 9 suc dom 𝑓 = (dom 𝑓 ∪ {dom 𝑓})
50 peano2 4595 . . . . . . . . 9 (dom 𝑓 ∈ ω → suc dom 𝑓 ∈ ω)
5149, 50eqeltrrid 2265 . . . . . . . 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 elomssom 4605 . . . . . . 7 ((dom 𝑓 ∪ {dom 𝑓}) ∈ ω → (dom 𝑓 ∪ {dom 𝑓}) ⊆ ω)
5452, 53syl 14 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (dom 𝑓 ∪ {dom 𝑓}) ⊆ ω)
55 elpm2r 6666 . . . . . 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))
5625, 26, 48, 54, 55syl22anc 1239 . . . . 5 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ (𝐴pm ω))
5748fdmd 5373 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) = (dom 𝑓 ∪ {dom 𝑓}))
5857, 52eqeltrd 2254 . . . . 5 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ ω)
5920, 56, 58elrabd 2896 . . . 4 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω})
60 ennnfonelemh.dceq . . . . . 6 (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
6160adantr 276 . . . . 5 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
6261, 42, 17ennnfonelemdc 12400 . . . 4 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → DECID (𝐹𝑗) ∈ (𝐹𝑗))
6318, 59, 62ifcldadc 3564 . . 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})
642, 13, 16, 17, 63ovmpod 6002 . 2 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) = if((𝐹𝑗) ∈ (𝐹𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩})))
6564, 63eqeltrd 2254 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 104  DECID wdc 834   = wceq 1353  wcel 2148  wne 2347  wral 2455  wrex 2456  {crab 2459  Vcvv 2738  cun 3128  wss 3130  c0 3423  ifcif 3535  {csn 3593  cop 3596  cmpt 4065  Ord word 4363  suc csuc 4366  ωcom 4590  ccnv 4626  dom cdm 4627  cima 4630  wf 5213  ontowfo 5215  cfv 5217  (class class class)co 5875  cmpo 5877  freccfrec 6391  pm cpm 6649  0cc0 7811  1c1 7812   + caddc 7814  cmin 8128  0cn0 9176  cz 9253  seqcseq 10445
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-coll 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  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-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-pm 6651
This theorem is referenced by:  ennnfonelemh  12405  ennnfonelem0  12406  ennnfonelemp1  12407  ennnfonelemom  12409
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