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Theorem ennnfonelemh 13155

Description: Lemma for ennnfone 13176. (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 13152	. . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
9	1, 2, 3, 4, 5, 6, 7	ennnfonelemg 13154	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
10		nn0uz 9889	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
11		0zd 9589	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
12	1, 2, 3, 4, 5, 6, 7	ennnfonelemjn 13153	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ_≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω)
13	8, 9, 10, 11, 12	seqf2 10830	. . 3 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ₀⟶{𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
14		ssrab2 3323	. . . 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 5522	. 2 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ₀⟶(𝐴 ↑_pm ω))
17	7	feq1i 5501	. 2 ⊢ (𝐻:ℕ₀⟶(𝐴 ↑_pm ω) ↔ seq0(𝐺, 𝐽):ℕ₀⟶(𝐴 ↑_pm ω))
18	16, 17	sylibr 134	1 ⊢ (𝜑 → 𝐻:ℕ₀⟶(𝐴 ↑_pm ω))

Colors of variables: wff set class

Syntax hints: → wi 4 DECID wdc 842 = wceq 1398 ∈ wcel 2203 ≠ wne 2412 ∀wral 2520 ∃wrex 2521 {crab 2524 ∪ cun 3209 ⊆ wss 3211 ∅c0 3508 ifcif 3620 {csn 3689 ⟨cop 3692 ↦ cmpt 4171 suc csuc 4486 ωcom 4712 ^◡ccnv 4748 dom cdm 4749 “ cima 4752 ⟶wf 5348 –onto→wfo 5350 ‘cfv 5352 (class class class)co 6050 ∈ cmpo 6052 freccfrec 6621 ↑_pm cpm 6883 0cc0 8127 1c1 8128 + caddc 8130 − cmin 8444 ℕ₀cn0 9496 ℤcz 9577 seqcseq 10809

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-addcom 8227 ax-addass 8229 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-0id 8235 ax-rnegex 8236 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-ltadd 8243

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-pm 6885 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-inn 9238 df-n0 9497 df-z 9578 df-uz 9854 df-seqfrec 10810

This theorem is referenced by: ennnfonelemp1 13157 ennnfonelemrnh 13167 ennnfonelemfun 13168 ennnfonelemf1 13169

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