Theorem ennnfonelemrn 12352

Description: Lemma for ennnfone 12358. 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 12351	. . 3 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)
10		f1f 5393	. . 3 ⊢ (𝐿:dom 𝐿–1-1→𝐴 → 𝐿:dom 𝐿⟶𝐴)
11		frn 5346	. . 3 ⊢ (𝐿:dom 𝐿⟶𝐴 → ran 𝐿 ⊆ 𝐴)
12	9, 10, 11	3syl 17	. 2 ⊢ (𝜑 → ran 𝐿 ⊆ 𝐴)
13		foelrn 5721	. . . . . 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 9203	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 0 ∈ ℤ)
16		simprl 521	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑗 ∈ ω)
17		peano2 4572	. . . . . . . . 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 10337	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝑁‘suc 𝑗) ∈ (ℤ≥‘0))
20		nn0uz 9500	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
21	19, 20	eleqtrrdi 2260	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝑁‘suc 𝑗) ∈ ℕ0)
22		fofn 5412	. . . . . . . . . 10 ⊢ (𝐹:ω–onto→𝐴 → 𝐹 Fn ω)
23	2, 22	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn ω)
24	23	ad2antrr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝐹 Fn ω)
25		ordom 4584	. . . . . . . . 9 ⊢ Ord ω
26		ordsucss 4481	. . . . . . . . 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 2729	. . . . . . . . . 10 ⊢ 𝑗 ∈ V
29	28	sucid 4395	. . . . . . . . 9 ⊢ 𝑗 ∈ suc 𝑗
30	29	a1i 9	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑗 ∈ suc 𝑗)
31		fnfvima 5719	. . . . . . . 8 ⊢ ((𝐹 Fn ω ∧ suc 𝑗 ⊆ ω ∧ 𝑗 ∈ suc 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ suc 𝑗))
32	24, 27, 30, 31	syl3anc 1228	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝐹‘𝑗) ∈ (𝐹 “ suc 𝑗))
33		simprr 522	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑤 = (𝐹‘𝑗))
34	15, 5	frec2uzf1od 10341	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑁:ω–1-1-onto→(ℤ≥‘0))
35		f1ocnvfv1 5745	. . . . . . . . 9 ⊢ ((𝑁:ω–1-1-onto→(ℤ≥‘0) ∧ suc 𝑗 ∈ ω) → (◡𝑁‘(𝑁‘suc 𝑗)) = suc 𝑗)
36	34, 18, 35	syl2anc 409	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (◡𝑁‘(𝑁‘suc 𝑗)) = suc 𝑗)
37	36	imaeq2d 4946	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))) = (𝐹 “ suc 𝑗))
38	32, 33, 37	3eltr4d 2250	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))))
39		fveq2 5486	. . . . . . . . 9 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (◡𝑁‘𝑖) = (◡𝑁‘(𝑁‘suc 𝑗)))
40	39	imaeq2d 4946	. . . . . . . 8 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (𝐹 “ (◡𝑁‘𝑖)) = (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))))
41	40	eleq2d 2236	. . . . . . 7 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)) ↔ 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗)))))
42	41	rspcev 2830	. . . . . 6 ⊢ (((𝑁‘suc 𝑗) ∈ ℕ0 ∧ 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗)))) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
43	21, 38, 42	syl2anc 409	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
44	14, 43	rexlimddv 2588	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
45		eliun 3870	. . . 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 4832	. . . . . . 7 ⊢ ran 𝐿 = ran ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)
48		rniun 5014	. . . . . . 7 ⊢ ran ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) = ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖)
49	47, 48	eqtri 2186	. . . . . 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 12346	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝐻‘𝑖):dom (𝐻‘𝑖)–1-1-onto→(𝐹 “ (◡𝑁‘𝑖)))
55		f1ofo 5439	. . . . . . . 8 ⊢ ((𝐻‘𝑖):dom (𝐻‘𝑖)–1-1-onto→(𝐹 “ (◡𝑁‘𝑖)) → (𝐻‘𝑖):dom (𝐻‘𝑖)–onto→(𝐹 “ (◡𝑁‘𝑖)))
56		forn 5413	. . . . . . . 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 3887	. . . . . 6 ⊢ (𝜑 → ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖) = ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
59	49, 58	syl5eq 2211	. . . . 5 ⊢ (𝜑 → ran 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
60	59	eleq2d 2236	. . . 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 3161	1 ⊢ (𝜑 → ran 𝐿 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 824   = wceq 1343   ∈ wcel 2136   ≠ wne 2336  ∀wral 2444  ∃wrex 2445   ∪ cun 3114   ⊆ wss 3116  ∅c0 3409  ifcif 3520  {csn 3576  ⟨cop 3579  ∪ ciun 3866   ↦ cmpt 4043  Ord word 4340  suc csuc 4343  ωcom 4567  ^◡ccnv 4603  dom cdm 4604  ran crn 4605   “ cima 4607   Fn wfn 5183  ⟶wf 5184  –1-1→wf1 5185  –onto→wfo 5186  –1-1-onto→wf1o 5187  ‘cfv 5188  (class class class)co 5842   ∈ cmpo 5844  freccfrec 6358   ↑_pm cpm 6615  0cc0 7753  1c1 7754   + caddc 7756   − cmin 8069  ℕ₀cn0 9114  ℤcz 9191  ℤ_≥cuz 9466  seqcseq 10380

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pm 6617  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-seqfrec 10381

This theorem is referenced by:  ennnfonelemen  12354