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Theorem ennnfonelemh 12775

Description: Lemma for ennnfone 12796. (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 12772	. . . 4 ⊢ (𝜑 → (𝐽‘0) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
9	1, 2, 3, 4, 5, 6, 7	ennnfonelemg 12774	. . . 4 ⊢ ((𝜑 ∧ (𝑓 ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω} ∧ 𝑗 ∈ ω)) → (𝑓𝐺𝑗) ∈ {𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
10		nn0uz 9683	. . . 4 ⊢ ℕ₀ = (ℤ_≥‘0)
11		0zd 9384	. . . 4 ⊢ (𝜑 → 0 ∈ ℤ)
12	1, 2, 3, 4, 5, 6, 7	ennnfonelemjn 12773	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (ℤ_≥‘(0 + 1))) → (𝐽‘𝑓) ∈ ω)
13	8, 9, 10, 11, 12	seqf2 10613	. . 3 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ₀⟶{𝑔 ∈ (𝐴 ↑_pm ω) ∣ dom 𝑔 ∈ ω})
14		ssrab2 3278	. . . 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 5438	. 2 ⊢ (𝜑 → seq0(𝐺, 𝐽):ℕ₀⟶(𝐴 ↑_pm ω))
17	7	feq1i 5418	. 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 836 = wceq 1373 ∈ wcel 2176 ≠ wne 2376 ∀wral 2484 ∃wrex 2485 {crab 2488 ∪ cun 3164 ⊆ wss 3166 ∅c0 3460 ifcif 3571 {csn 3633 ⟨cop 3636 ↦ cmpt 4105 suc csuc 4412 ωcom 4638 ^◡ccnv 4674 dom cdm 4675 “ cima 4678 ⟶wf 5267 –onto→wfo 5269 ‘cfv 5271 (class class class)co 5944 ∈ cmpo 5946 freccfrec 6476 ↑_pm cpm 6736 0cc0 7925 1c1 7926 + caddc 7928 − cmin 8243 ℕ₀cn0 9295 ℤcz 9372 seqcseq 10592

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 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-coll 4159 ax-sep 4162 ax-nul 4170 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-iinf 4636 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-addcom 8025 ax-addass 8027 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-0id 8033 ax-rnegex 8034 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-ltadd 8041

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-csb 3094 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-if 3572 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-iun 3929 df-br 4045 df-opab 4106 df-mpt 4107 df-tr 4143 df-id 4340 df-iord 4413 df-on 4415 df-ilim 4416 df-suc 4418 df-iom 4639 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-ima 4688 df-iota 5232 df-fun 5273 df-fn 5274 df-f 5275 df-f1 5276 df-fo 5277 df-f1o 5278 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-1st 6226 df-2nd 6227 df-recs 6391 df-frec 6477 df-pm 6738 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-inn 9037 df-n0 9296 df-z 9373 df-uz 9649 df-seqfrec 10593

This theorem is referenced by: ennnfonelemp1 12777 ennnfonelemrnh 12787 ennnfonelemfun 12788 ennnfonelemf1 12789

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