Theorem ennnfonelemf1 12373

Description: Lemma for ennnfone 12380. 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 12372	. . . 4 ⊢ (𝜑 → Fun 𝐿)
10	9	funfnd 5229	. . 3 ⊢ (𝜑 → 𝐿 Fn dom 𝐿)
11	1, 2, 3, 4, 5, 6, 7	ennnfonelemh 12359	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
12	11	ffnd 5348	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn ℕ0)
13		fniunfv 5741	. . . . . . . 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	eqtrid 2215	. . . . . 6 ⊢ (𝜑 → 𝐿 = ∪ ran 𝐻)
16	15	rneqd 4840	. . . . 5 ⊢ (𝜑 → ran 𝐿 = ran ∪ ran 𝐻)
17		rnuni 5022	. . . . 5 ⊢ ran ∪ ran 𝐻 = ∪ 𝑥 ∈ ran 𝐻ran 𝑥
18	16, 17	eqtrdi 2219	. . . 4 ⊢ (𝜑 → ran 𝐿 = ∪ 𝑥 ∈ ran 𝐻ran 𝑥)
19	11	frnd 5357	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐴 ↑pm ω))
20	19	sselda 3147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → 𝑥 ∈ (𝐴 ↑pm ω))
21		elpmi 6645	. . . . . . . . 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 5357	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → ran 𝑥 ⊆ 𝐴)
25	24	ralrimiva 2543	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐻ran 𝑥 ⊆ 𝐴)
26		iunss 3914	. . . . 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 3183	. . 3 ⊢ (𝜑 → ran 𝐿 ⊆ 𝐴)
29		df-f 5202	. . 3 ⊢ (𝐿:dom 𝐿⟶𝐴 ↔ (𝐿 Fn dom 𝐿 ∧ ran 𝐿 ⊆ 𝐴))
30	10, 28, 29	sylanbrc 415	. 2 ⊢ (𝜑 → 𝐿:dom 𝐿⟶𝐴)
31	19	sselda 3147	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → 𝑠 ∈ (𝐴 ↑pm ω))
32		pmfun 6646	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴 ↑pm ω) → Fun 𝑠)
33	31, 32	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → Fun 𝑠)
34	11	ffund 5351	. . . . . . . . . 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 5635	. . . . . . . . 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 5354	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐻 = ℕ0)
43	42	eleq2d 2240	. . . . . . . . . . . . 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 12368	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → (𝐻‘𝑞):dom (𝐻‘𝑞)–1-1-onto→(𝐹 “ (◡𝑁‘𝑞)))
46		f1ocnv 5455	. . . . . . . . . . 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 5444	. . . . . . . . . . 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 506	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → Fun ◡(𝐻‘𝑞))
50		simprr 527	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → 𝑠 = (𝐻‘𝑞))
51	50	cnveqd 4787	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → ◡𝑠 = ◡(𝐻‘𝑞))
52	51	funeqd 5220	. . . . . . . . 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 2592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → Fun ◡𝑠)
55	1	ad2antrr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
56	2	ad2antrr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝐹:ω–onto→𝐴)
57	3	ad2antrr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
58		simplr 525	. . . . . . . . 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 12371	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → (𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠))
61	60	ralrimiva 2543	. . . . . . 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 2543	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ ran 𝐻((Fun 𝑠 ∧ Fun ◡𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠)))
64		fun11uni 5268	. . . . 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 4787	. . . 4 ⊢ (𝜑 → ◡𝐿 = ◡∪ ran 𝐻)
68	67	funeqd 5220	. . 3 ⊢ (𝜑 → (Fun ◡𝐿 ↔ Fun ◡∪ ran 𝐻))
69	66, 68	mpbird 166	. 2 ⊢ (𝜑 → Fun ◡𝐿)
70		df-f1 5203	. 2 ⊢ (𝐿:dom 𝐿–1-1→𝐴 ↔ (𝐿:dom 𝐿⟶𝐴 ∧ Fun ◡𝐿))
71	30, 69, 70	sylanbrc 415	1 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 703  DECID wdc 829   = wceq 1348   ∈ wcel 2141   ≠ wne 2340  ∀wral 2448  ∃wrex 2449   ∪ cun 3119   ⊆ wss 3121  ∅c0 3414  ifcif 3526  {csn 3583  ⟨cop 3586  ∪ cuni 3796  ∪ ciun 3873   ↦ cmpt 4050  suc csuc 4350  ωcom 4574  ^◡ccnv 4610  dom cdm 4611  ran crn 4612   “ cima 4614  Fun wfun 5192   Fn wfn 5193  ⟶wf 5194  –1-1→wf1 5195  –onto→wfo 5196  –1-1-onto→wf1o 5197  ‘cfv 5198  (class class class)co 5853   ∈ cmpo 5855  freccfrec 6369   ↑_pm cpm 6627  0cc0 7774  1c1 7775   + caddc 7777   − cmin 8090  ℕ₀cn0 9135  ℤcz 9212  seqcseq 10401

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pm 6629  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-seqfrec 10402

This theorem is referenced by:  ennnfonelemrn  12374  ennnfonelemen  12376