Theorem ennnfonelemrn 12636

Description: Lemma for ennnfone 12642. 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 12635	. . 3 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)
10		f1f 5463	. . 3 ⊢ (𝐿:dom 𝐿–1-1→𝐴 → 𝐿:dom 𝐿⟶𝐴)
11		frn 5416	. . 3 ⊢ (𝐿:dom 𝐿⟶𝐴 → ran 𝐿 ⊆ 𝐴)
12	9, 10, 11	3syl 17	. 2 ⊢ (𝜑 → ran 𝐿 ⊆ 𝐴)
13		foelrn 5799	. . . . . 6 ⊢ ((𝐹:ω–onto→𝐴 ∧ 𝑤 ∈ 𝐴) → ∃𝑗 ∈ ω 𝑤 = (𝐹‘𝑗))
14	2, 13	sylan 283	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∃𝑗 ∈ ω 𝑤 = (𝐹‘𝑗))
15		0zd 9338	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 0 ∈ ℤ)
16		simprl 529	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑗 ∈ ω)
17		peano2 4631	. . . . . . . . 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 10494	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝑁‘suc 𝑗) ∈ (ℤ≥‘0))
20		nn0uz 9636	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
21	19, 20	eleqtrrdi 2290	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝑁‘suc 𝑗) ∈ ℕ0)
22		fofn 5482	. . . . . . . . . 10 ⊢ (𝐹:ω–onto→𝐴 → 𝐹 Fn ω)
23	2, 22	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn ω)
24	23	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝐹 Fn ω)
25		ordom 4643	. . . . . . . . 9 ⊢ Ord ω
26		ordsucss 4540	. . . . . . . . 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 2766	. . . . . . . . . 10 ⊢ 𝑗 ∈ V
29	28	sucid 4452	. . . . . . . . 9 ⊢ 𝑗 ∈ suc 𝑗
30	29	a1i 9	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑗 ∈ suc 𝑗)
31		fnfvima 5797	. . . . . . . 8 ⊢ ((𝐹 Fn ω ∧ suc 𝑗 ⊆ ω ∧ 𝑗 ∈ suc 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ suc 𝑗))
32	24, 27, 30, 31	syl3anc 1249	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝐹‘𝑗) ∈ (𝐹 “ suc 𝑗))
33		simprr 531	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑤 = (𝐹‘𝑗))
34	15, 5	frec2uzf1od 10498	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑁:ω–1-1-onto→(ℤ≥‘0))
35		f1ocnvfv1 5824	. . . . . . . . 9 ⊢ ((𝑁:ω–1-1-onto→(ℤ≥‘0) ∧ suc 𝑗 ∈ ω) → (◡𝑁‘(𝑁‘suc 𝑗)) = suc 𝑗)
36	34, 18, 35	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (◡𝑁‘(𝑁‘suc 𝑗)) = suc 𝑗)
37	36	imaeq2d 5009	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))) = (𝐹 “ suc 𝑗))
38	32, 33, 37	3eltr4d 2280	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))))
39		fveq2 5558	. . . . . . . . 9 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (◡𝑁‘𝑖) = (◡𝑁‘(𝑁‘suc 𝑗)))
40	39	imaeq2d 5009	. . . . . . . 8 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (𝐹 “ (◡𝑁‘𝑖)) = (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))))
41	40	eleq2d 2266	. . . . . . 7 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)) ↔ 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗)))))
42	41	rspcev 2868	. . . . . 6 ⊢ (((𝑁‘suc 𝑗) ∈ ℕ0 ∧ 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗)))) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
43	21, 38, 42	syl2anc 411	. . . . 5 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
44	14, 43	rexlimddv 2619	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
45		eliun 3920	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)) ↔ ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
46	44, 45	sylibr 134	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
47	8	rneqi 4894	. . . . . . 7 ⊢ ran 𝐿 = ran ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)
48		rniun 5080	. . . . . . 7 ⊢ ran ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) = ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖)
49	47, 48	eqtri 2217	. . . . . 6 ⊢ ran 𝐿 = ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖)
50	1	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
51	2	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝐹:ω–onto→𝐴)
52	3	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
53		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0)
54	50, 51, 52, 4, 5, 6, 7, 53	ennnfonelemhf1o 12630	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝐻‘𝑖):dom (𝐻‘𝑖)–1-1-onto→(𝐹 “ (◡𝑁‘𝑖)))
55		f1ofo 5511	. . . . . . . 8 ⊢ ((𝐻‘𝑖):dom (𝐻‘𝑖)–1-1-onto→(𝐹 “ (◡𝑁‘𝑖)) → (𝐻‘𝑖):dom (𝐻‘𝑖)–onto→(𝐹 “ (◡𝑁‘𝑖)))
56		forn 5483	. . . . . . . 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 3937	. . . . . 6 ⊢ (𝜑 → ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖) = ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
59	49, 58	eqtrid 2241	. . . . 5 ⊢ (𝜑 → ran 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
60	59	eleq2d 2266	. . . 4 ⊢ (𝜑 → (𝑤 ∈ ran 𝐿 ↔ 𝑤 ∈ ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖))))
61	60	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝑤 ∈ ran 𝐿 ↔ 𝑤 ∈ ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖))))
62	46, 61	mpbird 167	. 2 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐿)
63	12, 62	eqelssd 3202	1 ⊢ (𝜑 → ran 𝐿 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  ∃wrex 2476   ∪ cun 3155   ⊆ wss 3157  ∅c0 3450  ifcif 3561  {csn 3622  ⟨cop 3625  ∪ ciun 3916   ↦ cmpt 4094  Ord word 4397  suc csuc 4400  ωcom 4626  ^◡ccnv 4662  dom cdm 4663  ran crn 4664   “ cima 4666   Fn wfn 5253  ⟶wf 5254  –1-1→wf1 5255  –onto→wfo 5256  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922   ∈ cmpo 5924  freccfrec 6448   ↑_pm cpm 6708  0cc0 7879  1c1 7880   + caddc 7882   − cmin 8197  ℕ₀cn0 9249  ℤcz 9326  ℤ_≥cuz 9601  seqcseq 10539

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pm 6710  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-seqfrec 10540

This theorem is referenced by:  ennnfonelemen  12638