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Theorem ennnfonelemen 12581

Description: Lemma for ennnfone 12585. 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 12578	. . . . . 6 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)
10	1, 2, 3, 4, 5, 6, 7, 8	ennnfonelemdm 12580	. . . . . . 7 ⊢ (𝜑 → dom 𝐿 = ω)
11		f1eq2 5456	. . . . . . 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 12579	. . . . 5 ⊢ (𝜑 → ran 𝐿 = 𝐴)
15		dff1o5 5510	. . . . 5 ⊢ (𝐿:ω–1-1-onto→𝐴 ↔ (𝐿:ω–1-1→𝐴 ∧ ran 𝐿 = 𝐴))
16	13, 14, 15	sylanbrc 417	. . . 4 ⊢ (𝜑 → 𝐿:ω–1-1-onto→𝐴)
17		omex 4626	. . . . 5 ⊢ ω ∈ V
18	17	f1oen 6815	. . . 4 ⊢ (𝐿:ω–1-1-onto→𝐴 → ω ≈ 𝐴)
19	16, 18	syl 14	. . 3 ⊢ (𝜑 → ω ≈ 𝐴)
20	19	ensymd 6839	. 2 ⊢ (𝜑 → 𝐴 ≈ ω)
21		nnenom 10508	. . 3 ⊢ ℕ ≈ ω
22	21	ensymi 6838	. 2 ⊢ ω ≈ ℕ
23		entr 6840	. 2 ⊢ ((𝐴 ≈ ω ∧ ω ≈ ℕ) → 𝐴 ≈ ℕ)
24	20, 22, 23	sylancl 413	1 ⊢ (𝜑 → 𝐴 ≈ ℕ)

Colors of variables: wff set class

Syntax hints: → wi 4 ↔ wb 105 DECID wdc 835 = wceq 1364 ∈ wcel 2164 ≠ wne 2364 ∀wral 2472 ∃wrex 2473 ∪ cun 3152 ∅c0 3447 ifcif 3558 {csn 3619 ⟨cop 3622 ∪ ciun 3913 class class class wbr 4030 ↦ cmpt 4091 suc csuc 4397 ωcom 4623 ^◡ccnv 4659 dom cdm 4660 ran crn 4661 “ cima 4663 –1-1→wf1 5252 –onto→wfo 5253 –1-1-onto→wf1o 5254 ‘cfv 5255 (class class class)co 5919 ∈ cmpo 5921 freccfrec 6445 ↑_pm cpm 6705 ≈ cen 6794 0cc0 7874 1c1 7875 + caddc 7877 − cmin 8192 ℕcn 8984 ℕ₀cn0 9243 ℤcz 9320 seqcseq 10521

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-addcom 7974 ax-addass 7976 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-0id 7982 ax-rnegex 7983 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-ltadd 7990

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-er 6589 df-pm 6707 df-en 6797 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-inn 8985 df-n0 9244 df-z 9321 df-uz 9596 df-seqfrec 10522

This theorem is referenced by: ennnfonelemnn0 12582

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