ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ennnfonelemg GIF version

Theorem ennnfonelemg 12453
Description: Lemma for ennnfone 12475. 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 5538 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝐹𝑦) = (𝐹𝑗))
53imaeq2d 4988 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝐹𝑦) = (𝐹𝑗))
64, 5eleq12d 2260 . . . . 5 ((𝑥 = 𝑓𝑦 = 𝑗) → ((𝐹𝑦) ∈ (𝐹𝑦) ↔ (𝐹𝑗) ∈ (𝐹𝑗)))
7 simpl 109 . . . . 5 ((𝑥 = 𝑓𝑦 = 𝑗) → 𝑥 = 𝑓)
87dmeqd 4847 . . . . . . . 8 ((𝑥 = 𝑓𝑦 = 𝑗) → dom 𝑥 = dom 𝑓)
98, 4opeq12d 3801 . . . . . . 7 ((𝑥 = 𝑓𝑦 = 𝑗) → ⟨dom 𝑥, (𝐹𝑦)⟩ = ⟨dom 𝑓, (𝐹𝑗)⟩)
109sneqd 3620 . . . . . 6 ((𝑥 = 𝑓𝑦 = 𝑗) → {⟨dom 𝑥, (𝐹𝑦)⟩} = {⟨dom 𝑓, (𝐹𝑗)⟩})
117, 10uneq12d 3305 . . . . 5 ((𝑥 = 𝑓𝑦 = 𝑗) → (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩}) = (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}))
126, 7, 11ifbieq12d 3575 . . . 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 3255 . . . 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 3168 . . 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 4845 . . . . . 6 (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) → dom 𝑔 = dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}))
2019eleq1d 2258 . . . . 5 (𝑔 = (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) → (dom 𝑔 ∈ ω ↔ dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ ω))
21 omex 4610 . . . . . . . 8 ω ∈ V
22 ennnfonelemh.f . . . . . . . 8 (𝜑𝐹:ω–onto𝐴)
23 focdmex 6139 . . . . . . . 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 2905 . . . . . . . . . 10 (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} → 𝑓 ∈ (𝐴pm ω))
29 elpmi 6692 . . . . . . . . . 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 4845 . . . . . . . . . . 11 (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
3433eleq1d 2258 . . . . . . . . . 10 (𝑔 = 𝑓 → (dom 𝑔 ∈ ω ↔ dom 𝑓 ∈ ω))
3534elrab 2908 . . . . . . . . 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 4629 . . . . . . . . 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 4559 . . . . . . . 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 5457 . . . . . . . . . 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 5672 . . . . . . . 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 5736 . . . . . . 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 1254 . . . . . 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 4389 . . . . . . . . 9 suc dom 𝑓 = (dom 𝑓 ∪ {dom 𝑓})
50 peano2 4612 . . . . . . . . 9 (dom 𝑓 ∈ ω → suc dom 𝑓 ∈ ω)
5149, 50eqeltrrid 2277 . . . . . . . 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 4622 . . . . . . 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 6691 . . . . . 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 1250 . . . . 5 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ (𝐴pm ω))
5748fdmd 5391 . . . . . 6 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) = (dom 𝑓 ∪ {dom 𝑓}))
5857, 52eqeltrd 2266 . . . . 5 (((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) ∧ ¬ (𝐹𝑗) ∈ (𝐹𝑗)) → dom (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩}) ∈ ω)
5920, 56, 58elrabd 2910 . . . 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 12449 . . . 4 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → DECID (𝐹𝑗) ∈ (𝐹𝑗))
6318, 59, 62ifcldadc 3578 . . 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 6023 . 2 ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) = if((𝐹𝑗) ∈ (𝐹𝑗), 𝑓, (𝑓 ∪ {⟨dom 𝑓, (𝐹𝑗)⟩})))
6564, 63eqeltrd 2266 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 835   = wceq 1364  wcel 2160  wne 2360  wral 2468  wrex 2469  {crab 2472  Vcvv 2752  cun 3142  wss 3144  c0 3437  ifcif 3549  {csn 3607  cop 3610  cmpt 4079  Ord word 4380  suc csuc 4383  ωcom 4607  ccnv 4643  dom cdm 4644  cima 4647  wf 5231  ontowfo 5233  cfv 5235  (class class class)co 5895  cmpo 5897  freccfrec 6414  pm cpm 6674  0cc0 7840  1c1 7841   + caddc 7843  cmin 8157  0cn0 9205  cz 9282  seqcseq 10475
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-pm 6676
This theorem is referenced by:  ennnfonelemh  12454  ennnfonelem0  12455  ennnfonelemp1  12456  ennnfonelemom  12458
  Copyright terms: Public domain W3C validator