Theorem ennnfonelemrn 11932

Description: Lemma for ennnfone 11938. 𝐿 is onto 𝐴. (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
ennnfonelemrn	⊢ (𝜑 → ran 𝐿 = 𝐴)

Distinct variable groups:   𝐴,𝑗,𝑥,𝑦   𝑖,𝐹,𝑗,𝑥,𝑦,𝑘   𝑛,𝐹,𝑘   𝑗,𝐺   𝑖,𝐻,𝑗,𝑥,𝑦,𝑘   𝑗,𝐽   𝑖,𝑁,𝑗,𝑥,𝑦,𝑘   𝜑,𝑖,𝑗,𝑥,𝑦,𝑘   𝑗,𝑛

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

Proof of Theorem ennnfonelemrn

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemh.dceq	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
2		ennnfonelemh.f	. . . 4 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
3		ennnfonelemh.ne	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
4		ennnfonelemh.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
5		ennnfonelemh.n	. . . 4 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
6		ennnfonelemh.j	. . . 4 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
7		ennnfonelemh.h	. . . 4 ⊢ 𝐻 = seq0(𝐺, 𝐽)
8		ennnfone.l	. . . 4 ⊢ 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)
9	1, 2, 3, 4, 5, 6, 7, 8	ennnfonelemf1 11931	. . 3 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)
10		f1f 5328	. . 3 ⊢ (𝐿:dom 𝐿–1-1→𝐴 → 𝐿:dom 𝐿⟶𝐴)
11		frn 5281	. . 3 ⊢ (𝐿:dom 𝐿⟶𝐴 → ran 𝐿 ⊆ 𝐴)
12	9, 10, 11	3syl 17	. 2 ⊢ (𝜑 → ran 𝐿 ⊆ 𝐴)
13		foelrn 5654	. . . . . 6 ⊢ ((𝐹:ω–onto→𝐴 ∧ 𝑤 ∈ 𝐴) → ∃𝑗 ∈ ω 𝑤 = (𝐹‘𝑗))
14	2, 13	sylan 281	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∃𝑗 ∈ ω 𝑤 = (𝐹‘𝑗))
15		0zd 9066	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 0 ∈ ℤ)
16		simprl 520	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑗 ∈ ω)
17		peano2 4509	. . . . . . . . 9 ⊢ (𝑗 ∈ ω → suc 𝑗 ∈ ω)
18	16, 17	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → suc 𝑗 ∈ ω)
19	15, 5, 18	frec2uzuzd 10175	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝑁‘suc 𝑗) ∈ (ℤ≥‘0))
20		nn0uz 9360	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
21	19, 20	eleqtrrdi 2233	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝑁‘suc 𝑗) ∈ ℕ0)
22		fofn 5347	. . . . . . . . . 10 ⊢ (𝐹:ω–onto→𝐴 → 𝐹 Fn ω)
23	2, 22	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn ω)
24	23	ad2antrr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝐹 Fn ω)
25		ordom 4520	. . . . . . . . 9 ⊢ Ord ω
26		ordsucss 4420	. . . . . . . . 9 ⊢ (Ord ω → (𝑗 ∈ ω → suc 𝑗 ⊆ ω))
27	25, 16, 26	mpsyl 65	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → suc 𝑗 ⊆ ω)
28		vex 2689	. . . . . . . . . 10 ⊢ 𝑗 ∈ V
29	28	sucid 4339	. . . . . . . . 9 ⊢ 𝑗 ∈ suc 𝑗
30	29	a1i 9	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑗 ∈ suc 𝑗)
31		fnfvima 5652	. . . . . . . 8 ⊢ ((𝐹 Fn ω ∧ suc 𝑗 ⊆ ω ∧ 𝑗 ∈ suc 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ suc 𝑗))
32	24, 27, 30, 31	syl3anc 1216	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝐹‘𝑗) ∈ (𝐹 “ suc 𝑗))
33		simprr 521	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑤 = (𝐹‘𝑗))
34	15, 5	frec2uzf1od 10179	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑁:ω–1-1-onto→(ℤ≥‘0))
35		f1ocnvfv1 5678	. . . . . . . . 9 ⊢ ((𝑁:ω–1-1-onto→(ℤ≥‘0) ∧ suc 𝑗 ∈ ω) → (◡𝑁‘(𝑁‘suc 𝑗)) = suc 𝑗)
36	34, 18, 35	syl2anc 408	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (◡𝑁‘(𝑁‘suc 𝑗)) = suc 𝑗)
37	36	imaeq2d 4881	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))) = (𝐹 “ suc 𝑗))
38	32, 33, 37	3eltr4d 2223	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))))
39		fveq2 5421	. . . . . . . . 9 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (◡𝑁‘𝑖) = (◡𝑁‘(𝑁‘suc 𝑗)))
40	39	imaeq2d 4881	. . . . . . . 8 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (𝐹 “ (◡𝑁‘𝑖)) = (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))))
41	40	eleq2d 2209	. . . . . . 7 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)) ↔ 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗)))))
42	41	rspcev 2789	. . . . . 6 ⊢ (((𝑁‘suc 𝑗) ∈ ℕ0 ∧ 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗)))) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
43	21, 38, 42	syl2anc 408	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
44	14, 43	rexlimddv 2554	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
45		eliun 3817	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)) ↔ ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
46	44, 45	sylibr 133	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
47	8	rneqi 4767	. . . . . . 7 ⊢ ran 𝐿 = ran ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)
48		rniun 4949	. . . . . . 7 ⊢ ran ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) = ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖)
49	47, 48	eqtri 2160	. . . . . 6 ⊢ ran 𝐿 = ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖)
50	1	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
51	2	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐹:ω–onto→𝐴)
52	3	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
53		simpr 109	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0)
54	50, 51, 52, 4, 5, 6, 7, 53	ennnfonelemhf1o 11926	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝐻‘𝑖):dom (𝐻‘𝑖)–1-1-onto→(𝐹 “ (◡𝑁‘𝑖)))
55		f1ofo 5374	. . . . . . . 8 ⊢ ((𝐻‘𝑖):dom (𝐻‘𝑖)–1-1-onto→(𝐹 “ (◡𝑁‘𝑖)) → (𝐻‘𝑖):dom (𝐻‘𝑖)–onto→(𝐹 “ (◡𝑁‘𝑖)))
56		forn 5348	. . . . . . . 8 ⊢ ((𝐻‘𝑖):dom (𝐻‘𝑖)–onto→(𝐹 “ (◡𝑁‘𝑖)) → ran (𝐻‘𝑖) = (𝐹 “ (◡𝑁‘𝑖)))
57	54, 55, 56	3syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ran (𝐻‘𝑖) = (𝐹 “ (◡𝑁‘𝑖)))
58	57	iuneq2dv 3834	. . . . . 6 ⊢ (𝜑 → ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖) = ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
59	49, 58	syl5eq 2184	. . . . 5 ⊢ (𝜑 → ran 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
60	59	eleq2d 2209	. . . 4 ⊢ (𝜑 → (𝑤 ∈ ran 𝐿 ↔ 𝑤 ∈ ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖))))
61	60	adantr 274	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ ran 𝐿 ↔ 𝑤 ∈ ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖))))
62	46, 61	mpbird 166	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐿)
63	12, 62	eqelssd 3116	1 ⊢ (𝜑 → ran 𝐿 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 819   = wceq 1331   ∈ wcel 1480   ≠ wne 2308  ∀wral 2416  ∃wrex 2417   ∪ cun 3069   ⊆ wss 3071  ∅c0 3363  ifcif 3474  {csn 3527  ⟨cop 3530  ∪ ciun 3813   ↦ cmpt 3989  Ord word 4284  suc csuc 4287  ωcom 4504  ^◡ccnv 4538  dom cdm 4539  ran crn 4540   “ cima 4542   Fn wfn 5118  ⟶wf 5119  –1-1→wf1 5120  –onto→wfo 5121  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774   ∈ cmpo 5776  freccfrec 6287   ↑_pm cpm 6543  0cc0 7620  1c1 7621   + caddc 7623   − cmin 7933  ℕ₀cn0 8977  ℤcz 9054  ℤ_≥cuz 9326  seqcseq 10218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-ltadd 7736

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pm 6545  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-inn 8721  df-n0 8978  df-z 9055  df-uz 9327  df-seqfrec 10219

This theorem is referenced by:  ennnfonelemen  11934