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Theorem ennnfonelemen 13312

Description: Lemma for ennnfone 13316. The result. (Contributed by Jim Kingdon, 16-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)
ennnfone.l	⊢ 𝐿 = ∪ 𝑖 ∈ ℕ₀ (𝐻‘𝑖)

**Assertion**
Ref	Expression
ennnfonelemen	⊢ (𝜑 → 𝐴 ≈ ℕ)

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

**Proof of Theorem ennnfonelemen**
Step	Hyp	Ref	Expression
1		ennnfonelemh.dceq	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
2		ennnfonelemh.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
3		ennnfonelemh.ne	. . . . . . 7 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
4		ennnfonelemh.g	. . . . . . 7 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑_pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
5		ennnfonelemh.n	. . . . . . 7 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
6		ennnfonelemh.j	. . . . . . 7 ⊢ 𝐽 = (𝑥 ∈ ℕ₀ ↦ if(𝑥 = 0, ∅, (^◡𝑁‘(𝑥 − 1))))
7		ennnfonelemh.h	. . . . . . 7 ⊢ 𝐻 = seq0(𝐺, 𝐽)
8		ennnfone.l	. . . . . . 7 ⊢ 𝐿 = ∪ 𝑖 ∈ ℕ₀ (𝐻‘𝑖)
9	1, 2, 3, 4, 5, 6, 7, 8	ennnfonelemf1 13309	. . . . . 6 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)
10	1, 2, 3, 4, 5, 6, 7, 8	ennnfonelemdm 13311	. . . . . . 7 ⊢ (𝜑 → dom 𝐿 = ω)
11		f1eq2 5594	. . . . . . 7 ⊢ (dom 𝐿 = ω → (𝐿:dom 𝐿–1-1→𝐴 ↔ 𝐿:ω–1-1→𝐴))
12	10, 11	syl 14	. . . . . 6 ⊢ (𝜑 → (𝐿:dom 𝐿–1-1→𝐴 ↔ 𝐿:ω–1-1→𝐴))
13	9, 12	mpbid 147	. . . . 5 ⊢ (𝜑 → 𝐿:ω–1-1→𝐴)
14	1, 2, 3, 4, 5, 6, 7, 8	ennnfonelemrn 13310	. . . . 5 ⊢ (𝜑 → ran 𝐿 = 𝐴)
15		dff1o5 5648	. . . . 5 ⊢ (𝐿:ω–1-1-onto→𝐴 ↔ (𝐿:ω–1-1→𝐴 ∧ ran 𝐿 = 𝐴))
16	13, 14, 15	sylanbrc 421	. . . 4 ⊢ (𝜑 → 𝐿:ω–1-1-onto→𝐴)
17		omex 4740	. . . . 5 ⊢ ω ∈ V
18	17	f1oen 7045	. . . 4 ⊢ (𝐿:ω–1-1-onto→𝐴 → ω ≈ 𝐴)
19	16, 18	syl 14	. . 3 ⊢ (𝜑 → ω ≈ 𝐴)
20	19	ensymd 7070	. 2 ⊢ (𝜑 → 𝐴 ≈ ω)
21		nnenom 10871	. . 3 ⊢ ℕ ≈ ω
22	21	ensymi 7069	. 2 ⊢ ω ≈ ℕ
23		entr 7071	. 2 ⊢ ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ)
24	20, 22, 23	sylancl 417	1 ⊢ (𝜑 → 𝐴 ≈ ℕ)

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ↔ wb 105 DECID wdc 846 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 ∀wral 2528 ∃wrex 2529 ∪ cun 3218 ∅c0 3520 ifcif 3638 {csn 3709 ⟨cop 3712 ∪ ciun 4012 class class class wbr 4130 ↦ cmpt 4192 suc csuc 4510 ωcom 4737 ^◡ccnv 4773 dom cdm 4774 ran crn 4775 “ cima 4777 –1-1→wf1 5374 –onto→wfo 5375 –1-1-onto→wf1o 5376 ‘cfv 5377 (class class class)co 6085 ∈ cmpo 6087 freccfrec 6661 ↑_pm cpm 6923 ≈ cen 7020 0cc0 8179 1c1 8180 + caddc 8182 − cmin 8497 ℕcn 9304 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-er 6807 df-pm 6925 df-en 7023 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: ennnfonelemnn0 13313

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