ennnfonelemh - Intuitionistic Logic Explorer

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Theorem ennnfonelemh 12092

Description: Lemma for ennnfone 12113. (Contributed by Jim Kingdon, 8-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
ennnfonelemh	⊢ (𝜑 → 𝐻:ℕ₀⟶(𝐴 ↑_pm ω))

Distinct variable groups: 𝐴,𝑗,𝑥,𝑦 𝑥,𝐹,𝑦 𝑗,𝐺 𝑗,𝐽 𝑥,𝑁 𝜑,𝑗,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑘,𝑛) 𝐴(𝑘,𝑛) 𝐹(𝑗,𝑘,𝑛) 𝐺(𝑥,𝑦,𝑘,𝑛) 𝐻(𝑥,𝑦,𝑗,𝑘,𝑛) 𝐽(𝑥,𝑦,𝑘,𝑛) 𝑁(𝑦,𝑗,𝑘,𝑛)

Proof of Theorem ennnfonelemh Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemh.dceq	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
2		ennnfonelemh.f	. . . . 5 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
3		ennnfonelemh.ne	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
4		ennnfonelemh.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑_pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
5		ennnfonelemh.n	. . . . 5 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
6		ennnfonelemh.j	. . . . 5 ⊢ 𝐽 = (𝑥 ∈ ℕ₀ ↦ if(𝑥 = 0, ∅, (^◡𝑁‘(𝑥 − 1))))
7		ennnfonelemh.h	. . . . 5 ⊢ 𝐻 = seq0(𝐺, 𝐽)
8	1, 2, 3, 4, 5, 6, 7	ennnfonelemj0 12089	. . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
9	1, 2, 3, 4, 5, 6, 7	ennnfonelemg 12091	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
10		nn0uz 9452	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
11		0zd 9158	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
12	1, 2, 3, 4, 5, 6, 7	ennnfonelemjn 12090	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ_≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω)
13	8, 9, 10, 11, 12	seqf2 10341	. . 3 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ₀⟶{𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
14		ssrab2 3209	. . . 4 ⊢ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ⊆ (𝐴 ↑_pm ω)
15	14	a1i 9	. . 3 ⊢ (𝜑 → {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ⊆ (𝐴 ↑_pm ω))
16	13, 15	fssd 5325	. 2 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ₀⟶(𝐴 ↑_pm ω))
17	7	feq1i 5305	. 2 ⊢ (𝐻:ℕ₀⟶(𝐴 ↑_pm ω) ↔ seq0(𝐺, 𝐽):ℕ₀⟶(𝐴 ↑_pm ω))
18	16, 17	sylibr 133	1 ⊢ (𝜑 → 𝐻:ℕ₀⟶(𝐴 ↑_pm ω))

Colors of variables: wff set class

Syntax hints: → wi 4 DECID wdc 820 = wceq 1332 ∈ wcel 2125 ≠ wne 2324 ∀wral 2432 ∃wrex 2433 {crab 2436 ∪ cun 3096 ⊆ wss 3098 ∅c0 3390 ifcif 3501 {csn 3556 ⟨cop 3559 ↦ cmpt 4021 suc csuc 4320 ωcom 4543 ^◡ccnv 4578 dom cdm 4579 “ cima 4582 ⟶wf 5159 –onto→wfo 5161 ‘cfv 5163 (class class class)co 5814 ∈ cmpo 5816 freccfrec 6327 ↑_pm cpm 6583 0cc0 7711 1c1 7712 + caddc 7714 − cmin 8025 ℕ₀cn0 9069 ℤcz 9146 seqcseq 10322

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-addcom 7811 ax-addass 7813 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-0id 7819 ax-rnegex 7820 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-ltadd 7827

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-1st 6078 df-2nd 6079 df-recs 6242 df-frec 6328 df-pm 6585 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-inn 8813 df-n0 9070 df-z 9147 df-uz 9419 df-seqfrec 10323

This theorem is referenced by: ennnfonelemp1 12094 ennnfonelemrnh 12104 ennnfonelemfun 12105 ennnfonelemf1 12106

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