Theorem ennnfonelemrn 12419

Description: Lemma for ennnfone 12425. 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 12418	. . 3 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)
10		f1f 5421	. . 3 ⊢ (𝐿:dom 𝐿–1-1→𝐴 → 𝐿:dom 𝐿⟶𝐴)
11		frn 5374	. . 3 ⊢ (𝐿:dom 𝐿⟶𝐴 → ran 𝐿 ⊆ 𝐴)
12	9, 10, 11	3syl 17	. 2 ⊢ (𝜑 → ran 𝐿 ⊆ 𝐴)
13		foelrn 5753	. . . . . 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 9264	. . . . . . . 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 4594	. . . . . . . . 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 10401	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝑁‘suc 𝑗) ∈ (ℤ≥‘0))
20		nn0uz 9561	. . . . . . 7 ⊢ ℕ0 = (ℤ≥‘0)
21	19, 20	eleqtrrdi 2271	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝑁‘suc 𝑗) ∈ ℕ0)
22		fofn 5440	. . . . . . . . . 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 4606	. . . . . . . . 9 ⊢ Ord ω
26		ordsucss 4503	. . . . . . . . 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 2740	. . . . . . . . . 10 ⊢ 𝑗 ∈ V
29	28	sucid 4417	. . . . . . . . 9 ⊢ 𝑗 ∈ suc 𝑗
30	29	a1i 9	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑗 ∈ suc 𝑗)
31		fnfvima 5751	. . . . . . . 8 ⊢ ((𝐹 Fn ω ∧ suc 𝑗 ⊆ ω ∧ 𝑗 ∈ suc 𝑗) → (𝐹‘𝑗) ∈ (𝐹 “ suc 𝑗))
32	24, 27, 30, 31	syl3anc 1238	. . . . . . 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 10405	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑁:ω–1-1-onto→(ℤ≥‘0))
35		f1ocnvfv1 5777	. . . . . . . . 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 4970	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))) = (𝐹 “ suc 𝑗))
38	32, 33, 37	3eltr4d 2261	. . . . . 6 ⊢ (((𝜑 ∧ 𝑤 ∈ 𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹‘𝑗))) → 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))))
39		fveq2 5515	. . . . . . . . 9 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (◡𝑁‘𝑖) = (◡𝑁‘(𝑁‘suc 𝑗)))
40	39	imaeq2d 4970	. . . . . . . 8 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (𝐹 “ (◡𝑁‘𝑖)) = (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗))))
41	40	eleq2d 2247	. . . . . . 7 ⊢ (𝑖 = (𝑁‘suc 𝑗) → (𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)) ↔ 𝑤 ∈ (𝐹 “ (◡𝑁‘(𝑁‘suc 𝑗)))))
42	41	rspcev 2841	. . . . . 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 2599	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (◡𝑁‘𝑖)))
45		eliun 3890	. . . 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 4855	. . . . . . 7 ⊢ ran 𝐿 = ran ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)
48		rniun 5039	. . . . . . 7 ⊢ ran ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) = ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖)
49	47, 48	eqtri 2198	. . . . . 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 12413	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ ℕ0) → (𝐻‘𝑖):dom (𝐻‘𝑖)–1-1-onto→(𝐹 “ (◡𝑁‘𝑖)))
55		f1ofo 5468	. . . . . . . 8 ⊢ ((𝐻‘𝑖):dom (𝐻‘𝑖)–1-1-onto→(𝐹 “ (◡𝑁‘𝑖)) → (𝐻‘𝑖):dom (𝐻‘𝑖)–onto→(𝐹 “ (◡𝑁‘𝑖)))
56		forn 5441	. . . . . . . 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 3907	. . . . . 6 ⊢ (𝜑 → ∪ 𝑖 ∈ ℕ0 ran (𝐻‘𝑖) = ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
59	49, 58	eqtrid 2222	. . . . 5 ⊢ (𝜑 → ran 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐹 “ (◡𝑁‘𝑖)))
60	59	eleq2d 2247	. . . 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 3174	1 ⊢ (𝜑 → ran 𝐿 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 834   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  ∃wrex 2456   ∪ cun 3127   ⊆ wss 3129  ∅c0 3422  ifcif 3534  {csn 3592  ⟨cop 3595  ∪ ciun 3886   ↦ cmpt 4064  Ord word 4362  suc csuc 4365  ωcom 4589  ^◡ccnv 4625  dom cdm 4626  ran crn 4627   “ cima 4629   Fn wfn 5211  ⟶wf 5212  –1-1→wf1 5213  –onto→wfo 5214  –1-1-onto→wf1o 5215  ‘cfv 5216  (class class class)co 5874   ∈ cmpo 5876  freccfrec 6390   ↑_pm cpm 6648  0cc0 7810  1c1 7811   + caddc 7813   − cmin 8127  ℕ₀cn0 9175  ℤcz 9252  ℤ_≥cuz 9527  seqcseq 10444

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pm 6650  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-inn 8919  df-n0 9176  df-z 9253  df-uz 9528  df-seqfrec 10445

This theorem is referenced by:  ennnfonelemen  12421