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Theorem ennnfonelemrn 12390
Description: Lemma for ennnfone 12396. 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 12389 . . 3 (𝜑𝐿:dom 𝐿1-1𝐴)
10 f1f 5416 . . 3 (𝐿:dom 𝐿1-1𝐴𝐿:dom 𝐿𝐴)
11 frn 5369 . . 3 (𝐿:dom 𝐿𝐴 → ran 𝐿𝐴)
129, 10, 113syl 17 . 2 (𝜑 → ran 𝐿𝐴)
13 foelrn 5747 . . . . . 6 ((𝐹:ω–onto𝐴𝑤𝐴) → ∃𝑗 ∈ ω 𝑤 = (𝐹𝑗))
142, 13sylan 283 . . . . 5 ((𝜑𝑤𝐴) → ∃𝑗 ∈ ω 𝑤 = (𝐹𝑗))
15 0zd 9241 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 0 ∈ ℤ)
16 simprl 529 . . . . . . . . 9 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑗 ∈ ω)
17 peano2 4590 . . . . . . . . 9 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
1816, 17syl 14 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → suc 𝑗 ∈ ω)
1915, 5, 18frec2uzuzd 10375 . . . . . . 7 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → (𝑁‘suc 𝑗) ∈ (ℤ‘0))
20 nn0uz 9538 . . . . . . 7 0 = (ℤ‘0)
2119, 20eleqtrrdi 2271 . . . . . 6 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → (𝑁‘suc 𝑗) ∈ ℕ0)
22 fofn 5435 . . . . . . . . . 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 4602 . . . . . . . . 9 Ord ω
26 ordsucss 4499 . . . . . . . . 9 (Ord ω → (𝑗 ∈ ω → suc 𝑗 ⊆ ω))
2725, 16, 26mpsyl 65 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → suc 𝑗 ⊆ ω)
28 vex 2740 . . . . . . . . . 10 𝑗 ∈ V
2928sucid 4413 . . . . . . . . 9 𝑗 ∈ suc 𝑗
3029a1i 9 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑗 ∈ suc 𝑗)
31 fnfvima 5745 . . . . . . . 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 10379 . . . . . . . . 9 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑁:ω–1-1-onto→(ℤ‘0))
35 f1ocnvfv1 5771 . . . . . . . . 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 4965 . . . . . . 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 5510 . . . . . . . . 9 (𝑖 = (𝑁‘suc 𝑗) → (𝑁𝑖) = (𝑁‘(𝑁‘suc 𝑗)))
4039imaeq2d 4965 . . . . . . . 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 2841 . . . . . 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 3888 . . . 4 (𝑤 𝑖 ∈ ℕ0 (𝐹 “ (𝑁𝑖)) ↔ ∃𝑖 ∈ ℕ0 𝑤 ∈ (𝐹 “ (𝑁𝑖)))
4644, 45sylibr 134 . . 3 ((𝜑𝑤𝐴) → 𝑤 𝑖 ∈ ℕ0 (𝐹 “ (𝑁𝑖)))
478rneqi 4850 . . . . . . 7 ran 𝐿 = ran 𝑖 ∈ ℕ0 (𝐻𝑖)
48 rniun 5034 . . . . . . 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 12384 . . . . . . . 8 ((𝜑𝑖 ∈ ℕ0) → (𝐻𝑖):dom (𝐻𝑖)–1-1-onto→(𝐹 “ (𝑁𝑖)))
55 f1ofo 5463 . . . . . . . 8 ((𝐻𝑖):dom (𝐻𝑖)–1-1-onto→(𝐹 “ (𝑁𝑖)) → (𝐻𝑖):dom (𝐻𝑖)–onto→(𝐹 “ (𝑁𝑖)))
56 forn 5436 . . . . . . . 8 ((𝐻𝑖):dom (𝐻𝑖)–onto→(𝐹 “ (𝑁𝑖)) → ran (𝐻𝑖) = (𝐹 “ (𝑁𝑖)))
5754, 55, 563syl 17 . . . . . . 7 ((𝜑𝑖 ∈ ℕ0) → ran (𝐻𝑖) = (𝐹 “ (𝑁𝑖)))
5857iuneq2dv 3905 . . . . . 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 3174 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 3127  wss 3129  c0 3422  ifcif 3534  {csn 3591  cop 3594   ciun 3884  cmpt 4061  Ord word 4358  suc csuc 4361  ωcom 4585  ccnv 4621  dom cdm 4622  ran crn 4623  cima 4625   Fn wfn 5206  wf 5207  1-1wf1 5208  ontowfo 5209  1-1-ontowf1o 5210  cfv 5211  (class class class)co 5868  cmpo 5870  freccfrec 6384  pm cpm 6642  0cc0 7789  1c1 7790   + caddc 7792  cmin 8105  0cn0 9152  cz 9229  cuz 9504  seqcseq 10418
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 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905
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 4289  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-frec 6385  df-pm 6644  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-n0 9153  df-z 9230  df-uz 9505  df-seqfrec 10419
This theorem is referenced by:  ennnfonelemen  12392
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