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Theorem ennnfonelemg 12374
Description: Lemma for ennnfone 12396. 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 5514 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝐹𝑦) = (𝐹𝑗))
53imaeq2d 4965 . . . . . 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 4824 . . . . . . . 8 ((𝑥 = 𝑓𝑦 = 𝑗) → dom 𝑥 = dom 𝑓)
98, 4opeq12d 3784 . . . . . . 7 ((𝑥 = 𝑓𝑦 = 𝑗) → ⟨dom 𝑥, (𝐹𝑦)⟩ = ⟨dom 𝑓, (𝐹𝑗)⟩)
109sneqd 3604 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → {⟨dom 𝑥, (𝐹𝑦)⟩} = {⟨dom 𝑓, (𝐹𝑗)⟩})
117, 10uneq12d 3290 . . . . 5 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩}) = (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}))
126, 7, 11ifbieq12d 3560 . . . 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 3240 . . . 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 3153 . . 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 4822 . . . . . 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 4588 . . . . . . . 8 ω ∈ V
22 ennnfonelemh.f . . . . . . . 8 (𝜑𝐹:ω–onto𝐴)
23 focdmex 6109 . . . . . . . 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 2890 . . . . . . . . . 10 (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} → 𝑓 ∈ (𝐴pm ω))
29 elpmi 6660 . . . . . . . . . 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 4822 . . . . . . . . . . 11 (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
3433eleq1d 2246 . . . . . . . . . 10 (𝑔 = 𝑓 → (dom 𝑔 ∈ ω ↔ dom 𝑓 ∈ ω))
3534elrab 2893 . . . . . . . . 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 4607 . . . . . . . . 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 4537 . . . . . . . 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 5433 . . . . . . . . . 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 5647 . . . . . . . 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 5711 . . . . . . 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 4367 . . . . . . . . 9 suc dom 𝑓 = (dom 𝑓 ∪ {dom 𝑓})
50 peano2 4590 . . . . . . . . 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 4600 . . . . . . 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 6659 . . . . . 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 5367 . . . . . 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 2895 . . . 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 12370 . . . 4 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → DECID (𝐹𝑗) ∈ (𝐹𝑗))
6318, 59, 62ifcldadc 3563 . . 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 5995 . 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 2737  cun 3127  wss 3129  c0 3422  ifcif 3534  {csn 3591  cop 3594  cmpt 4061  Ord word 4358  suc csuc 4361  ωcom 4585  ccnv 4621  dom cdm 4622  cima 4625  wf 5207  ontowfo 5209  cfv 5211  (class class class)co 5868  cmpo 5870  freccfrec 6384  pm cpm 6642  0cc0 7789  1c1 7790   + caddc 7792  cmin 8105  0cn0 9152  cz 9229  seqcseq 10418
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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
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 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pm 6644
This theorem is referenced by:  ennnfonelemh  12375  ennnfonelem0  12376  ennnfonelemp1  12377  ennnfonelemom  12379
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