Theorem ennnfonelemf1 12660

Description: Lemma for ennnfone 12667. 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 12659	. . . 4 ⊢ (𝜑 → Fun 𝐿)
10	9	funfnd 5290	. . 3 ⊢ (𝜑 → 𝐿 Fn dom 𝐿)
11	1, 2, 3, 4, 5, 6, 7	ennnfonelemh 12646	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
12	11	ffnd 5411	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn ℕ0)
13		fniunfv 5812	. . . . . . . 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 2241	. . . . . 6 ⊢ (𝜑 → 𝐿 = ∪ ran 𝐻)
16	15	rneqd 4896	. . . . 5 ⊢ (𝜑 → ran 𝐿 = ran ∪ ran 𝐻)
17		rnuni 5082	. . . . 5 ⊢ ran ∪ ran 𝐻 = ∪ 𝑥 ∈ ran 𝐻ran 𝑥
18	16, 17	eqtrdi 2245	. . . 4 ⊢ (𝜑 → ran 𝐿 = ∪ 𝑥 ∈ ran 𝐻ran 𝑥)
19	11	frnd 5420	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐴 ↑pm ω))
20	19	sselda 3184	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → 𝑥 ∈ (𝐴 ↑pm ω))
21		elpmi 6735	. . . . . . . . 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 112	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → 𝑥:dom 𝑥⟶𝐴)
24	23	frnd 5420	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → ran 𝑥 ⊆ 𝐴)
25	24	ralrimiva 2570	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐻ran 𝑥 ⊆ 𝐴)
26		iunss 3958	. . . . 5 ⊢ (∪ 𝑥 ∈ ran 𝐻ran 𝑥 ⊆ 𝐴 ↔ ∀𝑥 ∈ ran 𝐻ran 𝑥 ⊆ 𝐴)
27	25, 26	sylibr 134	. . . 4 ⊢ (𝜑 → ∪ 𝑥 ∈ ran 𝐻ran 𝑥 ⊆ 𝐴)
28	18, 27	eqsstrd 3220	. . 3 ⊢ (𝜑 → ran 𝐿 ⊆ 𝐴)
29		df-f 5263	. . 3 ⊢ (𝐿:dom 𝐿⟶𝐴 ↔ (𝐿 Fn dom 𝐿 ∧ ran 𝐿 ⊆ 𝐴))
30	10, 28, 29	sylanbrc 417	. 2 ⊢ (𝜑 → 𝐿:dom 𝐿⟶𝐴)
31	19	sselda 3184	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → 𝑠 ∈ (𝐴 ↑pm ω))
32		pmfun 6736	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴 ↑pm ω) → Fun 𝑠)
33	31, 32	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → Fun 𝑠)
34	11	ffund 5414	. . . . . . . . . 10 ⊢ (𝜑 → Fun 𝐻)
35	34	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → Fun 𝐻)
36		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → 𝑠 ∈ ran 𝐻)
37		elrnrexdm 5704	. . . . . . . . 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 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
40	2	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → 𝐹:ω–onto→𝐴)
41	3	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
42	11	fdmd 5417	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐻 = ℕ0)
43	42	eleq2d 2266	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑞 ∈ dom 𝐻 ↔ 𝑞 ∈ ℕ0))
44	43	biimpa 296	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → 𝑞 ∈ ℕ0)
45	39, 40, 41, 4, 5, 6, 7, 44	ennnfonelemhf1o 12655	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → (𝐻‘𝑞):dom (𝐻‘𝑞)–1-1-onto→(𝐹 “ (◡𝑁‘𝑞)))
46		f1ocnv 5520	. . . . . . . . . . 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 5509	. . . . . . . . . . 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 509	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → Fun ◡(𝐻‘𝑞))
50		simprr 531	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → 𝑠 = (𝐻‘𝑞))
51	50	cnveqd 4843	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → ◡𝑠 = ◡(𝐻‘𝑞))
52	51	funeqd 5281	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → (Fun ◡𝑠 ↔ Fun ◡(𝐻‘𝑞)))
53	49, 52	mpbird 167	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → Fun ◡𝑠)
54	38, 53	rexlimddv 2619	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → Fun ◡𝑠)
55	1	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
56	2	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝐹:ω–onto→𝐴)
57	3	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
58		simplr 528	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝑠 ∈ ran 𝐻)
59		simpr 110	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → 𝑡 ∈ ran 𝐻)
60	55, 56, 57, 4, 5, 6, 7, 58, 59	ennnfonelemrnh 12658	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → (𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠))
61	60	ralrimiva 2570	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠))
62	33, 54, 61	jca31 309	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → ((Fun 𝑠 ∧ Fun ◡𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠)))
63	62	ralrimiva 2570	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ ran 𝐻((Fun 𝑠 ∧ Fun ◡𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠)))
64		fun11uni 5329	. . . . 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 114	. . 3 ⊢ (𝜑 → Fun ◡∪ ran 𝐻)
67	15	cnveqd 4843	. . . 4 ⊢ (𝜑 → ◡𝐿 = ◡∪ ran 𝐻)
68	67	funeqd 5281	. . 3 ⊢ (𝜑 → (Fun ◡𝐿 ↔ Fun ◡∪ ran 𝐻))
69	66, 68	mpbird 167	. 2 ⊢ (𝜑 → Fun ◡𝐿)
70		df-f1 5264	. 2 ⊢ (𝐿:dom 𝐿–1-1→𝐴 ↔ (𝐿:dom 𝐿⟶𝐴 ∧ Fun ◡𝐿))
71	30, 69, 70	sylanbrc 417	1 ⊢ (𝜑 → 𝐿:dom 𝐿–1-1→𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 709  DECID wdc 835   = wceq 1364   ∈ wcel 2167   ≠ wne 2367  ∀wral 2475  ∃wrex 2476   ∪ cun 3155   ⊆ wss 3157  ∅c0 3451  ifcif 3562  {csn 3623  ⟨cop 3626  ∪ cuni 3840  ∪ ciun 3917   ↦ cmpt 4095  suc csuc 4401  ωcom 4627  ^◡ccnv 4663  dom cdm 4664  ran crn 4665   “ cima 4667  Fun wfun 5253   Fn wfn 5254  ⟶wf 5255  –1-1→wf1 5256  –onto→wfo 5257  –1-1-onto→wf1o 5258  ‘cfv 5259  (class class class)co 5925   ∈ cmpo 5927  freccfrec 6457   ↑_pm cpm 6717  0cc0 7896  1c1 7897   + caddc 7899   − cmin 8214  ℕ₀cn0 9266  ℤcz 9343  seqcseq 10556

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 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-addass 7998  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-ltadd 8012

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 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-frec 6458  df-pm 6719  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-inn 9008  df-n0 9267  df-z 9344  df-uz 9619  df-seqfrec 10557

This theorem is referenced by:  ennnfonelemrn  12661  ennnfonelemen  12663