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Theorem ennnfonelemrn 12420
Description: Lemma for ennnfone 12426. 𝐿 is onto 𝐴. (Contributed by Jim Kingdon, 16-Jul-2023.)
Hypotheses
Ref Expression
ennnfonelemh.dceq (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f (𝜑𝐹:ω–onto𝐴)
ennnfonelemh.ne (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
ennnfonelemh.g 𝐺 = (𝑥 ∈ (𝐴pm ω), 𝑦 ∈ ω ↦ if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})))
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 (𝐻𝑖)
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.
StepHypRef Expression
1 ennnfonelemh.dceq . . . 4 (𝜑 → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
2 ennnfonelemh.f . . . 4 (𝜑𝐹:ω–onto𝐴)
3 ennnfonelemh.ne . . . 4 (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
4 ennnfonelemh.g . . . 4 𝐺 = (𝑥 ∈ (𝐴pm ω), 𝑦 ∈ ω ↦ if((𝐹𝑦) ∈ (𝐹𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹𝑦)⟩})))
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 (𝐻𝑖)
91, 2, 3, 4, 5, 6, 7, 8ennnfonelemf1 12419 . . 3 (𝜑𝐿:dom 𝐿1-1𝐴)
10 f1f 5422 . . 3 (𝐿:dom 𝐿1-1𝐴𝐿:dom 𝐿𝐴)
11 frn 5375 . . 3 (𝐿:dom 𝐿𝐴 → ran 𝐿𝐴)
129, 10, 113syl 17 . 2 (𝜑 → ran 𝐿𝐴)
13 foelrn 5754 . . . . . 6 ((𝐹:ω–onto𝐴𝑤𝐴) → ∃𝑗 ∈ ω 𝑤 = (𝐹𝑗))
142, 13sylan 283 . . . . 5 ((𝜑𝑤𝐴) → ∃𝑗 ∈ ω 𝑤 = (𝐹𝑗))
15 0zd 9265 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 0 ∈ ℤ)
16 simprl 529 . . . . . . . . 9 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑗 ∈ ω)
17 peano2 4595 . . . . . . . . 9 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
1816, 17syl 14 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → suc 𝑗 ∈ ω)
1915, 5, 18frec2uzuzd 10402 . . . . . . 7 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → (𝑁‘suc 𝑗) ∈ (ℤ‘0))
20 nn0uz 9562 . . . . . . 7 0 = (ℤ‘0)
2119, 20eleqtrrdi 2271 . . . . . 6 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → (𝑁‘suc 𝑗) ∈ ℕ0)
22 fofn 5441 . . . . . . . . . 10 (𝐹:ω–onto𝐴𝐹 Fn ω)
232, 22syl 14 . . . . . . . . 9 (𝜑𝐹 Fn ω)
2423ad2antrr 488 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝐹 Fn ω)
25 ordom 4607 . . . . . . . . 9 Ord ω
26 ordsucss 4504 . . . . . . . . 9 (Ord ω → (𝑗 ∈ ω → suc 𝑗 ⊆ ω))
2725, 16, 26mpsyl 65 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → suc 𝑗 ⊆ ω)
28 vex 2741 . . . . . . . . . 10 𝑗 ∈ V
2928sucid 4418 . . . . . . . . 9 𝑗 ∈ suc 𝑗
3029a1i 9 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑗 ∈ suc 𝑗)
31 fnfvima 5752 . . . . . . . 8 ((𝐹 Fn ω ∧ suc 𝑗 ⊆ ω ∧ 𝑗 ∈ suc 𝑗) → (𝐹𝑗) ∈ (𝐹 “ suc 𝑗))
3224, 27, 30, 31syl3anc 1238 . . . . . . 7 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → (𝐹𝑗) ∈ (𝐹 “ suc 𝑗))
33 simprr 531 . . . . . . 7 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑤 = (𝐹𝑗))
3415, 5frec2uzf1od 10406 . . . . . . . . 9 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑁:ω–1-1-onto→(ℤ‘0))
35 f1ocnvfv1 5778 . . . . . . . . 9 ((𝑁:ω–1-1-onto→(ℤ‘0) ∧ suc 𝑗 ∈ ω) → (𝑁‘(𝑁‘suc 𝑗)) = suc 𝑗)
3634, 18, 35syl2anc 411 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → (𝑁‘(𝑁‘suc 𝑗)) = suc 𝑗)
3736imaeq2d 4971 . . . . . . 7 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → (𝐹 “ (𝑁‘(𝑁‘suc 𝑗))) = (𝐹 “ suc 𝑗))
3832, 33, 373eltr4d 2261 . . . . . 6 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑤 ∈ (𝐹 “ (𝑁‘(𝑁‘suc 𝑗))))
39 fveq2 5516 . . . . . . . . 9 (𝑖 = (𝑁‘suc 𝑗) → (𝑁𝑖) = (𝑁‘(𝑁‘suc 𝑗)))
4039imaeq2d 4971 . . . . . . . 8 (𝑖 = (𝑁‘suc 𝑗) → (𝐹 “ (𝑁𝑖)) = (𝐹 “ (𝑁‘(𝑁‘suc 𝑗))))
4140eleq2d 2247 . . . . . . 7 (𝑖 = (𝑁‘suc 𝑗) → (𝑤 ∈ (𝐹 “ (𝑁𝑖)) ↔ 𝑤 ∈ (𝐹 “ (𝑁‘(𝑁‘suc 𝑗)))))
4241rspcev 2842 . . . . . 6 (((𝑁‘suc 𝑗) ∈ ℕ0𝑤 ∈ (𝐹 “ (𝑁‘(𝑁‘suc 𝑗)))) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (𝑁𝑖)))
4321, 38, 42syl2anc 411 . . . . 5 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (𝑁𝑖)))
4414, 43rexlimddv 2599 . . . 4 ((𝜑𝑤𝐴) → ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (𝑁𝑖)))
45 eliun 3891 . . . 4 (𝑤 𝑖 ∈ ℕ0 (𝐹 “ (𝑁𝑖)) ↔ ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (𝑁𝑖)))
4644, 45sylibr 134 . . 3 ((𝜑𝑤𝐴) → 𝑤 𝑖 ∈ ℕ0 (𝐹 “ (𝑁𝑖)))
478rneqi 4856 . . . . . . 7 ran 𝐿 = ran 𝑖 ∈ ℕ0 (𝐻𝑖)
48 rniun 5040 . . . . . . 7 ran 𝑖 ∈ ℕ0 (𝐻𝑖) = 𝑖 ∈ ℕ0 ran (𝐻𝑖)
4947, 48eqtri 2198 . . . . . 6 ran 𝐿 = 𝑖 ∈ ℕ0 ran (𝐻𝑖)
501adantr 276 . . . . . . . . 9 ((𝜑𝑖 ∈ ℕ0) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
512adantr 276 . . . . . . . . 9 ((𝜑𝑖 ∈ ℕ0) → 𝐹:ω–onto𝐴)
523adantr 276 . . . . . . . . 9 ((𝜑𝑖 ∈ ℕ0) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹𝑘) ≠ (𝐹𝑗))
53 simpr 110 . . . . . . . . 9 ((𝜑𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0)
5450, 51, 52, 4, 5, 6, 7, 53ennnfonelemhf1o 12414 . . . . . . . 8 ((𝜑𝑖 ∈ ℕ0) → (𝐻𝑖):dom (𝐻𝑖)–1-1-onto→(𝐹 “ (𝑁𝑖)))
55 f1ofo 5469 . . . . . . . 8 ((𝐻𝑖):dom (𝐻𝑖)–1-1-onto→(𝐹 “ (𝑁𝑖)) → (𝐻𝑖):dom (𝐻𝑖)–onto→(𝐹 “ (𝑁𝑖)))
56 forn 5442 . . . . . . . 8 ((𝐻𝑖):dom (𝐻𝑖)–onto→(𝐹 “ (𝑁𝑖)) → ran (𝐻𝑖) = (𝐹 “ (𝑁𝑖)))
5754, 55, 563syl 17 . . . . . . 7 ((𝜑𝑖 ∈ ℕ0) → ran (𝐻𝑖) = (𝐹 “ (𝑁𝑖)))
5857iuneq2dv 3908 . . . . . 6 (𝜑 𝑖 ∈ ℕ0 ran (𝐻𝑖) = 𝑖 ∈ ℕ0 (𝐹 “ (𝑁𝑖)))
5949, 58eqtrid 2222 . . . . 5 (𝜑 → ran 𝐿 = 𝑖 ∈ ℕ0 (𝐹 “ (𝑁𝑖)))
6059eleq2d 2247 . . . 4 (𝜑 → (𝑤 ∈ ran 𝐿𝑤 𝑖 ∈ ℕ0 (𝐹 “ (𝑁𝑖))))
6160adantr 276 . . 3 ((𝜑𝑤𝐴) → (𝑤 ∈ ran 𝐿𝑤 𝑖 ∈ ℕ0 (𝐹 “ (𝑁𝑖))))
6246, 61mpbird 167 . 2 ((𝜑𝑤𝐴) → 𝑤 ∈ ran 𝐿)
6312, 62eqelssd 3175 1 (𝜑 → ran 𝐿 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  DECID wdc 834   = wceq 1353  wcel 2148  wne 2347  wral 2455  wrex 2456  cun 3128  wss 3130  c0 3423  ifcif 3535  {csn 3593  cop 3596   ciun 3887  cmpt 4065  Ord word 4363  suc csuc 4366  ωcom 4590  ccnv 4626  dom cdm 4627  ran crn 4628  cima 4630   Fn wfn 5212  wf 5213  1-1wf1 5214  ontowfo 5215  1-1-ontowf1o 5216  cfv 5217  (class class class)co 5875  cmpo 5877  freccfrec 6391  pm cpm 6649  0cc0 7811  1c1 7812   + caddc 7814  cmin 8128  0cn0 9176  cz 9253  cuz 9528  seqcseq 10445
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 4119  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588  ax-cnex 7902  ax-resscn 7903  ax-1cn 7904  ax-1re 7905  ax-icn 7906  ax-addcl 7907  ax-addrcl 7908  ax-mulcl 7909  ax-addcom 7911  ax-addass 7913  ax-distr 7915  ax-i2m1 7916  ax-0lt1 7917  ax-0id 7919  ax-rnegex 7920  ax-cnre 7922  ax-pre-ltirr 7923  ax-pre-ltwlin 7924  ax-pre-lttrn 7925  ax-pre-ltadd 7927
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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-ilim 4370  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-riota 5831  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1st 6141  df-2nd 6142  df-recs 6306  df-frec 6392  df-pm 6651  df-pnf 7994  df-mnf 7995  df-xr 7996  df-ltxr 7997  df-le 7998  df-sub 8130  df-neg 8131  df-inn 8920  df-n0 9177  df-z 9254  df-uz 9529  df-seqfrec 10446
This theorem is referenced by:  ennnfonelemen  12422
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