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Theorem ennnfonelemh 13295

Description: Lemma for ennnfone 13316. (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 13292	. . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
9	1, 2, 3, 4, 5, 6, 7	ennnfonelemg 13294	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
10		nn0uz 9957	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
11		0zd 9656	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
12	1, 2, 3, 4, 5, 6, 7	ennnfonelemjn 13293	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ_≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω)
13	8, 9, 10, 11, 12	seqf2 10905	. . 3 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ₀⟶{𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
14		ssrab2 3333	. . . 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 5547	. 2 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ₀⟶(𝐴 ↑_pm ω))
17	7	feq1i 5526	. 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

This proof depends on syntax axioms: → wi 4 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∀wral 2528 ∃wrex 2529 {crab 2532 ∪ cun 3218 ⊆ wss 3220 ∅c0 3520 ifcif 3638 {csn 3709 ⟨cop 3712 ↦ cmpt 4192 suc csuc 4510 ωcom 4737 ^◡ccnv 4773 dom cdm 4774 “ cima 4777 ⟶wf 5373 –onto→wfo 5375 ‘cfv 5377 (class class class)co 6085 ∈ cmpo 6087 freccfrec 6661 ↑_pm cpm 6923 0cc0 8179 1c1 8180 + caddc 8182 − cmin 8497 ℕ₀cn0 9563 ℤcz 9644 seqcseq 10884

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-addass 8281 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-ltadd 8295

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-pm 6925 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-inn 9305 df-n0 9564 df-z 9645 df-uz 9922 df-seqfrec 10885

This theorem is used by: ennnfonelemp1 13297 ennnfonelemrnh 13307 ennnfonelemfun 13308 ennnfonelemf1 13309

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