Theorem ennnfonelemrn 12790

Description: Lemma for ennnfone 12796. 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 12789	. . 3 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)
10		f1f 5481	. . 3 ⊢ (𝐿:dom 𝐿–1-1→𝐴 → 𝐿:dom 𝐿⟶𝐴)
11		frn 5434	. . 3 ⊢ (𝐿:dom 𝐿⟶𝐴 → ran 𝐿 ⊆ 𝐴)
12	9, 10, 11	3syl 17	. 2 ⊢ (𝜑 → ran 𝐿 ⊆ 𝐴)
13		foelrn 5821	. . . . . 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 9384	. . . . . . . 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 4643	. . . . . . . . 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 10547	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝑁‘suc 𝑗) ∈ (ℤ≥‘0))
20		nn0uz 9683	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
21	19, 20	eleqtrrdi 2299	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝑁‘suc 𝑗) ∈ ℕ0)
22		fofn 5500	. . . . . . . . . 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 4655	. . . . . . . . 9 ⊢ Ord ω
26		ordsucss 4552	. . . . . . . . 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 2775	. . . . . . . . . 10 ⊢ 𝑗 ∈ V
29	28	sucid 4464	. . . . . . . . 9 ⊢ 𝑗 ∈ suc 𝑗
30	29	a1i 9	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑗 ∈ suc 𝑗)
31		fnfvima 5819	. . . . . . . 8 ⊢ ((𝐹 Fn ω ∧ suc 𝑗 ⊆ ω ∧ 𝑗 ∈ suc 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ suc 𝑗))
32	24, 27, 30, 31	syl3anc 1250	. . . . . . 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 10551	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑁:ω–1-1-onto→(ℤ≥‘0))
35		f1ocnvfv1 5846	. . . . . . . . 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 5022	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))) = (𝐹 “ suc 𝑗))
38	32, 33, 37	3eltr4d 2289	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))))
39		fveq2 5576	. . . . . . . . 9 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (◡𝑁‘𝑖) = (◡𝑁‘(𝑁‘suc 𝑗)))
40	39	imaeq2d 5022	. . . . . . . 8 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (𝐹 “ (◡𝑁‘𝑖)) = (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))))
41	40	eleq2d 2275	. . . . . . 7 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)) ↔ 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗)))))
42	41	rspcev 2877	. . . . . 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 2628	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
45		eliun 3931	. . . 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 4906	. . . . . . 7 ⊢ ran 𝐿 = ran ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)
48		rniun 5093	. . . . . . 7 ⊢ ran ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) = ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖)
49	47, 48	eqtri 2226	. . . . . 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 12784	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝐻‘𝑖):dom (𝐻‘𝑖)–1-1-onto→(𝐹 “ (◡𝑁‘𝑖)))
55		f1ofo 5529	. . . . . . . 8 ⊢ ((𝐻‘𝑖):dom (𝐻‘𝑖)–1-1-onto→(𝐹 “ (◡𝑁‘𝑖)) → (𝐻‘𝑖):dom (𝐻‘𝑖)–onto→(𝐹 “ (◡𝑁‘𝑖)))
56		forn 5501	. . . . . . . 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 3948	. . . . . 6 ⊢ (𝜑 → ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖) = ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
59	49, 58	eqtrid 2250	. . . . 5 ⊢ (𝜑 → ran 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
60	59	eleq2d 2275	. . . 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 3212	1 ⊢ (𝜑 → ran 𝐿 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 836   = wceq 1373   ∈ wcel 2176   ≠ wne 2376  ∀wral 2484  ∃wrex 2485   ∪ cun 3164   ⊆ wss 3166  ∅c0 3460  ifcif 3571  {csn 3633  ⟨cop 3636  ∪ ciun 3927   ↦ cmpt 4105  Ord word 4409  suc csuc 4412  ωcom 4638  ^◡ccnv 4674  dom cdm 4675  ran crn 4676   “ cima 4678   Fn wfn 5266  ⟶wf 5267  –1-1→wf1 5268  –onto→wfo 5269  –1-1-onto→wf1o 5270  ‘cfv 5271  (class class class)co 5944   ∈ cmpo 5946  freccfrec 6476   ↑_pm cpm 6736  0cc0 7925  1c1 7926   + caddc 7928   − cmin 8243  ℕ₀cn0 9295  ℤcz 9372  ℤ_≥cuz 9648  seqcseq 10592

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-ltadd 8041

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pm 6738  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-seqfrec 10593

This theorem is referenced by:  ennnfonelemen  12792