Theorem ennnfonelemf1 12351

Description: Lemma for ennnfone 12358. 𝐿 is one-to-one. (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
ennnfonelemf1	⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)

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

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

Proof of Theorem ennnfonelemf1

Dummy variables 𝑞 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemh.dceq	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
2		ennnfonelemh.f	. . . . 5 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
3		ennnfonelemh.ne	. . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
4		ennnfonelemh.g	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
5		ennnfonelemh.n	. . . . 5 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
6		ennnfonelemh.j	. . . . 5 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
7		ennnfonelemh.h	. . . . 5 ⊢ 𝐻 = seq0(𝐺, 𝐽)
8		ennnfone.l	. . . . 5 ⊢ 𝐿 = ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖)
9	1, 2, 3, 4, 5, 6, 7, 8	ennnfonelemfun 12350	. . . 4 ⊢ (𝜑 → Fun 𝐿)
10	9	funfnd 5219	. . 3 ⊢ (𝜑 → 𝐿 Fn dom 𝐿)
11	1, 2, 3, 4, 5, 6, 7	ennnfonelemh 12337	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
12	11	ffnd 5338	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn ℕ0)
13		fniunfv 5730	. . . . . . . 8 ⊢ (𝐻 Fn ℕ0 → ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) = ∪ ran 𝐻)
14	12, 13	syl 14	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑖 ∈ ℕ0 (𝐻‘𝑖) = ∪ ran 𝐻)
15	8, 14	syl5eq 2211	. . . . . 6 ⊢ (𝜑 → 𝐿 = ∪ ran 𝐻)
16	15	rneqd 4833	. . . . 5 ⊢ (𝜑 → ran 𝐿 = ran ∪ ran 𝐻)
17		rnuni 5015	. . . . 5 ⊢ ran ∪ ran 𝐻 = ∪ 𝑥 ∈ ran 𝐻ran 𝑥
18	16, 17	eqtrdi 2215	. . . 4 ⊢ (𝜑 → ran 𝐿 = ∪ 𝑥 ∈ ran 𝐻ran 𝑥)
19	11	frnd 5347	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐴 ↑pm ω))
20	19	sselda 3142	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → 𝑥 ∈ (𝐴 ↑pm ω))
21		elpmi 6633	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ↑pm ω) → (𝑥:dom 𝑥⟶𝐴 ∧ dom 𝑥 ⊆ ω))
22	20, 21	syl 14	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → (𝑥:dom 𝑥⟶𝐴 ∧ dom 𝑥 ⊆ ω))
23	22	simpld 111	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → 𝑥:dom 𝑥⟶𝐴)
24	23	frnd 5347	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → ran 𝑥 ⊆ 𝐴)
25	24	ralrimiva 2539	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐻ran 𝑥 ⊆ 𝐴)
26		iunss 3907	. . . . 5 ⊢ (∪ 𝑥 ∈ ran 𝐻ran 𝑥 ⊆ 𝐴 ↔ ∀𝑥 ∈ ran 𝐻ran 𝑥 ⊆ 𝐴)
27	25, 26	sylibr 133	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ran 𝐻ran 𝑥 ⊆ 𝐴)
28	18, 27	eqsstrd 3178	. . 3 ⊢ (𝜑 → ran 𝐿 ⊆ 𝐴)
29		df-f 5192	. . 3 ⊢ (𝐿:dom 𝐿⟶𝐴 ↔ (𝐿 Fn dom 𝐿 ∧ ran 𝐿 ⊆ 𝐴))
30	10, 28, 29	sylanbrc 414	. 2 ⊢ (𝜑 → 𝐿:dom 𝐿⟶𝐴)
31	19	sselda 3142	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → 𝑠 ∈ (𝐴 ↑pm ω))
32		pmfun 6634	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴 ↑pm ω) → Fun 𝑠)
33	31, 32	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → Fun 𝑠)
34	11	ffund 5341	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐻)
35	34	adantr 274	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → Fun 𝐻)
36		simpr 109	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → 𝑠 ∈ ran 𝐻)
37		elrnrexdm 5624	. . . . . . . . 9 ⊢ (Fun 𝐻 → (𝑠 ∈ ran 𝐻 → ∃𝑞 ∈ dom 𝐻 𝑠 = (𝐻‘𝑞)))
38	35, 36, 37	sylc 62	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → ∃𝑞 ∈ dom 𝐻 𝑠 = (𝐻‘𝑞))
39	1	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
40	2	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → 𝐹:ω–onto→𝐴)
41	3	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
42	11	fdmd 5344	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐻 = ℕ0)
43	42	eleq2d 2236	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑞 ∈ dom 𝐻 ↔ 𝑞 ∈ ℕ0))
44	43	biimpa 294	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → 𝑞 ∈ ℕ0)
45	39, 40, 41, 4, 5, 6, 7, 44	ennnfonelemhf1o 12346	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → (𝐻‘𝑞):dom (𝐻‘𝑞)–1-1-onto→(𝐹 “ (◡𝑁‘𝑞)))
46		f1ocnv 5445	. . . . . . . . . . 11 ⊢ ((𝐻‘𝑞):dom (𝐻‘𝑞)–1-1-onto→(𝐹 “ (◡𝑁‘𝑞)) → ◡(𝐻‘𝑞):(𝐹 “ (◡𝑁‘𝑞))–1-1-onto→dom (𝐻‘𝑞))
47		f1ofun 5434	. . . . . . . . . . 11 ⊢ (◡(𝐻‘𝑞):(𝐹 “ (◡𝑁‘𝑞))–1-1-onto→dom (𝐻‘𝑞) → Fun ◡(𝐻‘𝑞))
48	45, 46, 47	3syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → Fun ◡(𝐻‘𝑞))
49	48	ad2ant2r 501	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → Fun ◡(𝐻‘𝑞))
50		simprr 522	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → 𝑠 = (𝐻‘𝑞))
51	50	cnveqd 4780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → ◡𝑠 = ◡(𝐻‘𝑞))
52	51	funeqd 5210	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → (Fun ◡𝑠 ↔ Fun ◡(𝐻‘𝑞)))
53	49, 52	mpbird 166	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → Fun ◡𝑠)
54	38, 53	rexlimddv 2588	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → Fun ◡𝑠)
55	1	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
56	2	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝐹:ω–onto→𝐴)
57	3	ad2antrr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
58		simplr 520	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝑠 ∈ ran 𝐻)
59		simpr 109	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝑡 ∈ ran 𝐻)
60	55, 56, 57, 4, 5, 6, 7, 58, 59	ennnfonelemrnh 12349	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → (𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠))
61	60	ralrimiva 2539	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠))
62	33, 54, 61	jca31 307	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → ((Fun 𝑠 ∧ Fun ◡𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠)))
63	62	ralrimiva 2539	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ ran 𝐻((Fun 𝑠 ∧ Fun ◡𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠)))
64		fun11uni 5258	. . . . 5 ⊢ (∀𝑠 ∈ ran 𝐻((Fun 𝑠 ∧ Fun ◡𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠)) → (Fun ∪ ran 𝐻 ∧ Fun ◡∪ ran 𝐻))
65	63, 64	syl 14	. . . 4 ⊢ (𝜑 → (Fun ∪ ran 𝐻 ∧ Fun ◡∪ ran 𝐻))
66	65	simprd 113	. . 3 ⊢ (𝜑 → Fun ◡∪ ran 𝐻)
67	15	cnveqd 4780	. . . 4 ⊢ (𝜑 → ◡𝐿 = ◡∪ ran 𝐻)
68	67	funeqd 5210	. . 3 ⊢ (𝜑 → (Fun ◡𝐿 ↔ Fun ◡∪ ran 𝐻))
69	66, 68	mpbird 166	. 2 ⊢ (𝜑 → Fun ◡𝐿)
70		df-f1 5193	. 2 ⊢ (𝐿:dom 𝐿–1-1→𝐴 ↔ (𝐿:dom 𝐿⟶𝐴 ∧ Fun ◡𝐿))
71	30, 69, 70	sylanbrc 414	1 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698  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  ∪ cuni 3789  ∪ ciun 3866   ↦ cmpt 4043  suc csuc 4343  ωcom 4567  ^◡ccnv 4603  dom cdm 4604  ran crn 4605   “ cima 4607  Fun wfun 5182   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  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:  ennnfonelemrn  12352  ennnfonelemen  12354