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Theorem ennnfonelemrn 12403
Description: Lemma for ennnfone 12409. 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 12402 . . 3 (𝜑𝐿:dom 𝐿1-1𝐴)
10 f1f 5417 . . 3 (𝐿:dom 𝐿1-1𝐴𝐿:dom 𝐿𝐴)
11 frn 5370 . . 3 (𝐿:dom 𝐿𝐴 → ran 𝐿𝐴)
129, 10, 113syl 17 . 2 (𝜑 → ran 𝐿𝐴)
13 foelrn 5748 . . . . . 6 ((𝐹:ω–onto𝐴𝑤𝐴) → ∃𝑗 ∈ ω 𝑤 = (𝐹𝑗))
142, 13sylan 283 . . . . 5 ((𝜑𝑤𝐴) → ∃𝑗 ∈ ω 𝑤 = (𝐹𝑗))
15 0zd 9254 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 0 ∈ ℤ)
16 simprl 529 . . . . . . . . 9 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑗 ∈ ω)
17 peano2 4591 . . . . . . . . 9 (𝑗 ∈ ω → suc 𝑗 ∈ ω)
1816, 17syl 14 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → suc 𝑗 ∈ ω)
1915, 5, 18frec2uzuzd 10388 . . . . . . 7 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → (𝑁‘suc 𝑗) ∈ (ℤ‘0))
20 nn0uz 9551 . . . . . . 7 0 = (ℤ‘0)
2119, 20eleqtrrdi 2271 . . . . . 6 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → (𝑁‘suc 𝑗) ∈ ℕ0)
22 fofn 5436 . . . . . . . . . 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 4603 . . . . . . . . 9 Ord ω
26 ordsucss 4500 . . . . . . . . 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 4414 . . . . . . . . 9 𝑗 ∈ suc 𝑗
3029a1i 9 . . . . . . . 8 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑗 ∈ suc 𝑗)
31 fnfvima 5746 . . . . . . . 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 10392 . . . . . . . . 9 (((𝜑𝑤𝐴) ∧ (𝑗 ∈ ω ∧ 𝑤 = (𝐹𝑗))) → 𝑁:ω–1-1-onto→(ℤ‘0))
35 f1ocnvfv1 5772 . . . . . . . . 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 4966 . . . . . . 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 5511 . . . . . . . . 9 (𝑖 = (𝑁‘suc 𝑗) → (𝑁𝑖) = (𝑁‘(𝑁‘suc 𝑗)))
4039imaeq2d 4966 . . . . . . . 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 4851 . . . . . . 7 ran 𝐿 = ran 𝑖 ∈ ℕ0 (𝐻𝑖)
48 rniun 5035 . . . . . . 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 12397 . . . . . . . 8 ((𝜑𝑖 ∈ ℕ0) → (𝐻𝑖):dom (𝐻𝑖)–1-1-onto→(𝐹 “ (𝑁𝑖)))
55 f1ofo 5464 . . . . . . . 8 ((𝐻𝑖):dom (𝐻𝑖)–1-1-onto→(𝐹 “ (𝑁𝑖)) → (𝐻𝑖):dom (𝐻𝑖)–onto→(𝐹 “ (𝑁𝑖)))
56 forn 5437 . . . . . . . 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 4359  suc csuc 4362  ωcom 4586  ccnv 4622  dom cdm 4623  ran crn 4624  cima 4626   Fn wfn 5207  wf 5208  1-1wf1 5209  ontowfo 5210  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  cmpo 5871  freccfrec 6385  pm cpm 6643  0cc0 7802  1c1 7803   + caddc 7805  cmin 8118  0cn0 9165  cz 9242  cuz 9517  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:  ennnfonelemen  12405
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