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Theorem ennnfonelemen 12907

Description: Lemma for ennnfone 12911. 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 12904	. . . . . 6 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)
10	1, 2, 3, 4, 5, 6, 7, 8	ennnfonelemdm 12906	. . . . . . 7 ⊢ (𝜑 → dom 𝐿 = ω)
11		f1eq2 5499	. . . . . . 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 12905	. . . . 5 ⊢ (𝜑 → ran 𝐿 = 𝐴)
15		dff1o5 5553	. . . . 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 4659	. . . . 5 ⊢ ω ∈ V
18	17	f1oen 6873	. . . 4 ⊢ (𝐿:ω–1-1-onto→𝐴 → ω ≈ 𝐴)
19	16, 18	syl 14	. . 3 ⊢ (𝜑 → ω ≈ 𝐴)
20	19	ensymd 6898	. 2 ⊢ (𝜑 → 𝐴 ≈ ω)
21		nnenom 10616	. . 3 ⊢ ℕ ≈ ω
22	21	ensymi 6897	. 2 ⊢ ω ≈ ℕ
23		entr 6899	. 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 836 = wceq 1373 ∈ wcel 2178 ≠ wne 2378 ∀wral 2486 ∃wrex 2487 ∪ cun 3172 ∅c0 3468 ifcif 3579 {csn 3643 ⟨cop 3646 ∪ ciun 3941 class class class wbr 4059 ↦ cmpt 4121 suc csuc 4430 ωcom 4656 ^◡ccnv 4692 dom cdm 4693 ran crn 4694 “ cima 4696 –1-1→wf1 5287 –onto→wfo 5288 –1-1-onto→wf1o 5289 ‘cfv 5290 (class class class)co 5967 ∈ cmpo 5969 freccfrec 6499 ↑_pm cpm 6759 ≈ cen 6848 0cc0 7960 1c1 7961 + caddc 7963 − cmin 8278 ℕcn 9071 ℕ₀cn0 9330 ℤcz 9407 seqcseq 10629

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 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2180 ax-14 2181 ax-ext 2189 ax-coll 4175 ax-sep 4178 ax-nul 4186 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-setind 4603 ax-iinf 4654 ax-cnex 8051 ax-resscn 8052 ax-1cn 8053 ax-1re 8054 ax-icn 8055 ax-addcl 8056 ax-addrcl 8057 ax-mulcl 8058 ax-addcom 8060 ax-addass 8062 ax-distr 8064 ax-i2m1 8065 ax-0lt1 8066 ax-0id 8068 ax-rnegex 8069 ax-cnre 8071 ax-pre-ltirr 8072 ax-pre-ltwlin 8073 ax-pre-lttrn 8074 ax-pre-ltadd 8076

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 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-reu 2493 df-rab 2495 df-v 2778 df-sbc 3006 df-csb 3102 df-dif 3176 df-un 3178 df-in 3180 df-ss 3187 df-nul 3469 df-if 3580 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-int 3900 df-iun 3943 df-br 4060 df-opab 4122 df-mpt 4123 df-tr 4159 df-id 4358 df-iord 4431 df-on 4433 df-ilim 4434 df-suc 4436 df-iom 4657 df-xp 4699 df-rel 4700 df-cnv 4701 df-co 4702 df-dm 4703 df-rn 4704 df-res 4705 df-ima 4706 df-iota 5251 df-fun 5292 df-fn 5293 df-f 5294 df-f1 5295 df-fo 5296 df-f1o 5297 df-fv 5298 df-riota 5922 df-ov 5970 df-oprab 5971 df-mpo 5972 df-1st 6249 df-2nd 6250 df-recs 6414 df-frec 6500 df-er 6643 df-pm 6761 df-en 6851 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147 df-le 8148 df-sub 8280 df-neg 8281 df-inn 9072 df-n0 9331 df-z 9408 df-uz 9684 df-seqfrec 10630

This theorem is referenced by: ennnfonelemnn0 12908

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