Theorem ennnfonelemf1 12402

Description: Lemma for ennnfone 12409. 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 12401	. . . 4 ⊢ (𝜑 → Fun 𝐿)
10	9	funfnd 5243	. . 3 ⊢ (𝜑 → 𝐿 Fn dom 𝐿)
11	1, 2, 3, 4, 5, 6, 7	ennnfonelemh 12388	. . . . . . . . 9 ⊢ (𝜑 → 𝐻:ℕ0⟶(𝐴 ↑pm ω))
12	11	ffnd 5362	. . . . . . . 8 ⊢ (𝜑 → 𝐻 Fn ℕ0)
13		fniunfv 5757	. . . . . . . 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 2222	. . . . . 6 ⊢ (𝜑 → 𝐿 = ∪ ran 𝐻)
16	15	rneqd 4852	. . . . 5 ⊢ (𝜑 → ran 𝐿 = ran ∪ ran 𝐻)
17		rnuni 5036	. . . . 5 ⊢ ran ∪ ran 𝐻 = ∪ 𝑥 ∈ ran 𝐻ran 𝑥
18	16, 17	eqtrdi 2226	. . . 4 ⊢ (𝜑 → ran 𝐿 = ∪ 𝑥 ∈ ran 𝐻ran 𝑥)
19	11	frnd 5371	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝐻 ⊆ (𝐴 ↑pm ω))
20	19	sselda 3155	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → 𝑥 ∈ (𝐴 ↑pm ω))
21		elpmi 6661	. . . . . . . . 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 5371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐻) → ran 𝑥 ⊆ 𝐴)
25	24	ralrimiva 2550	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ran 𝐻ran 𝑥 ⊆ 𝐴)
26		iunss 3925	. . . . 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 3191	. . 3 ⊢ (𝜑 → ran 𝐿 ⊆ 𝐴)
29		df-f 5216	. . 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 3155	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → 𝑠 ∈ (𝐴 ↑pm ω))
32		pmfun 6662	. . . . . . . 8 ⊢ (𝑠 ∈ (𝐴 ↑pm ω) → Fun 𝑠)
33	31, 32	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ ran 𝐻) → Fun 𝑠)
34	11	ffund 5365	. . . . . . . . . 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 5651	. . . . . . . . 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 5368	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom 𝐻 = ℕ0)
43	42	eleq2d 2247	. . . . . . . . . . . . 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 12397	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑞 ∈ dom 𝐻) → (𝐻‘𝑞):dom (𝐻‘𝑞)–1-1-onto→(𝐹 “ (◡𝑁‘𝑞)))
46		f1ocnv 5470	. . . . . . . . . . 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 5459	. . . . . . . . . . 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 4799	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ (𝑞 ∈ dom 𝐻 ∧ 𝑠 = (𝐻‘𝑞))) → ◡𝑠 = ◡(𝐻‘𝑞))
52	51	funeqd 5234	. . . . . . . . 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 2599	. . . . . . 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 12400	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ ran 𝐻) ∧ 𝑡 ∈ ran 𝐻) → (𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠))
61	60	ralrimiva 2550	. . . . . . 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 2550	. . . . 5 ⊢ (𝜑 → ∀𝑠 ∈ ran 𝐻((Fun 𝑠 ∧ Fun ◡𝑠) ∧ ∀𝑡 ∈ ran 𝐻(𝑠 ⊆ 𝑡 ∨ 𝑡 ⊆ 𝑠)))
64		fun11uni 5282	. . . . 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 4799	. . . 4 ⊢ (𝜑 → ◡𝐿 = ◡∪ ran 𝐻)
68	67	funeqd 5234	. . 3 ⊢ (𝜑 → (Fun ◡𝐿 ↔ Fun ◡∪ ran 𝐻))
69	66, 68	mpbird 167	. 2 ⊢ (𝜑 → Fun ◡𝐿)
70		df-f1 5217	. 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 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148   ≠ wne 2347  ∀wral 2455  ∃wrex 2456   ∪ cun 3127   ⊆ wss 3129  ∅c0 3422  ifcif 3534  {csn 3591  ⟨cop 3594  ∪ cuni 3807  ∪ ciun 3884   ↦ cmpt 4061  suc csuc 4362  ωcom 4586  ^◡ccnv 4622  dom cdm 4623  ran crn 4624   “ cima 4626  Fun wfun 5206   Fn wfn 5207  ⟶wf 5208  –1-1→wf1 5209  –onto→wfo 5210  –1-1-onto→wf1o 5211  ‘cfv 5212  (class class class)co 5869   ∈ cmpo 5871  freccfrec 6385   ↑_pm cpm 6643  0cc0 7802  1c1 7803   + caddc 7805   − cmin 8118  ℕ₀cn0 9165  ℤcz 9242  seqcseq 10431

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 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918

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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pm 6645  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-seqfrec 10432

This theorem is referenced by:  ennnfonelemrn  12403  ennnfonelemen  12405