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Theorem isf32lem9 9221
Description: Lemma for isfin3-2 9227. Construction of the onto function. (Contributed by Stefan O'Rear, 5-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
isf32lem.a (𝜑𝐹:ω⟶𝒫 𝐺)
isf32lem.b (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
isf32lem.c (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)
isf32lem.d 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹𝑦)}
isf32lem.e 𝐽 = (𝑢 ∈ ω ↦ (𝑣𝑆 (𝑣𝑆) ≈ 𝑢))
isf32lem.f 𝐾 = ((𝑤𝑆 ↦ ((𝐹𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)
isf32lem.g 𝐿 = (𝑡𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))))
Assertion
Ref Expression
isf32lem9 (𝜑𝐿:𝐺onto→ω)
Distinct variable groups:   𝑥,𝑤   𝑡,𝐺   𝑥,𝐿   𝑡,𝑠,𝑢,𝑣,𝑤,𝑥,𝑦,𝜑   𝑤,𝐹,𝑥,𝑦   𝑆,𝑠,𝑡,𝑢,𝑣,𝑤,𝑥,𝑦   𝐽,𝑠,𝑡,𝑤,𝑥,𝑦   𝐾,𝑠,𝑡,𝑥,𝑦
Allowed substitution hints:   𝐹(𝑣,𝑢,𝑡,𝑠)   𝐺(𝑥,𝑦,𝑤,𝑣,𝑢,𝑠)   𝐽(𝑣,𝑢)   𝐾(𝑤,𝑣,𝑢)   𝐿(𝑦,𝑤,𝑣,𝑢,𝑡,𝑠)

Proof of Theorem isf32lem9
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isf32lem.g . . . 4 𝐿 = (𝑡𝐺 ↦ (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))))
2 ssab2 3719 . . . . . . 7 {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))} ⊆ ω
3 iotacl 5912 . . . . . . 7 (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))})
42, 3sseldi 3634 . . . . . 6 (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ ω)
5 iotanul 5904 . . . . . . 7 (¬ ∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) = ∅)
6 peano1 7127 . . . . . . 7 ∅ ∈ ω
75, 6syl6eqel 2738 . . . . . 6 (¬ ∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ ω)
84, 7pm2.61i 176 . . . . 5 (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ ω
98a1i 11 . . . 4 (𝑡𝐺 → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ ω)
101, 9fmpti 6423 . . 3 𝐿:𝐺⟶ω
1110a1i 11 . 2 (𝜑𝐿:𝐺⟶ω)
12 isf32lem.a . . . . . 6 (𝜑𝐹:ω⟶𝒫 𝐺)
13 isf32lem.b . . . . . 6 (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹𝑥))
14 isf32lem.c . . . . . 6 (𝜑 → ¬ ran 𝐹 ∈ ran 𝐹)
15 isf32lem.d . . . . . 6 𝑆 = {𝑦 ∈ ω ∣ (𝐹‘suc 𝑦) ⊊ (𝐹𝑦)}
16 isf32lem.e . . . . . 6 𝐽 = (𝑢 ∈ ω ↦ (𝑣𝑆 (𝑣𝑆) ≈ 𝑢))
17 isf32lem.f . . . . . 6 𝐾 = ((𝑤𝑆 ↦ ((𝐹𝑤) ∖ (𝐹‘suc 𝑤))) ∘ 𝐽)
1812, 13, 14, 15, 16, 17isf32lem6 9218 . . . . 5 ((𝜑𝑎 ∈ ω) → (𝐾𝑎) ≠ ∅)
19 n0 3964 . . . . 5 ((𝐾𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐾𝑎))
2018, 19sylib 208 . . . 4 ((𝜑𝑎 ∈ ω) → ∃𝑏 𝑏 ∈ (𝐾𝑎))
2112, 13, 14, 15, 16, 17isf32lem8 9220 . . . . . . . . 9 ((𝜑𝑎 ∈ ω) → (𝐾𝑎) ⊆ 𝐺)
2221sselda 3636 . . . . . . . 8 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → 𝑏𝐺)
23 eleq1 2718 . . . . . . . . . . . . 13 (𝑡 = 𝑏 → (𝑡 ∈ (𝐾𝑠) ↔ 𝑏 ∈ (𝐾𝑠)))
2423anbi2d 740 . . . . . . . . . . . 12 (𝑡 = 𝑏 → ((𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) ↔ (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
2524iotabidv 5910 . . . . . . . . . . 11 (𝑡 = 𝑏 → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
26 iotaex 5906 . . . . . . . . . . 11 (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ V
2725, 1, 26fvmpt3i 6326 . . . . . . . . . 10 (𝑏𝐺 → (𝐿𝑏) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
2822, 27syl 17 . . . . . . . . 9 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → (𝐿𝑏) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
29 simp1r 1106 . . . . . . . . . . . . . . . 16 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → 𝑏 ∈ (𝐾𝑎))
30 simpl1 1084 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝜑)
31 simpr 476 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑠𝑎)
3231necomd 2878 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑎𝑠)
33 simpl2 1085 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑎 ∈ ω)
34 simpl3 1086 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑠 ∈ ω)
3512, 13, 14, 15, 16, 17isf32lem7 9219 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎𝑠) ∧ (𝑎 ∈ ω ∧ 𝑠 ∈ ω)) → ((𝐾𝑎) ∩ (𝐾𝑠)) = ∅)
3630, 32, 33, 34, 35syl22anc 1367 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → ((𝐾𝑎) ∩ (𝐾𝑠)) = ∅)
37 disj1 4052 . . . . . . . . . . . . . . . . . . . . 21 (((𝐾𝑎) ∩ (𝐾𝑠)) = ∅ ↔ ∀𝑏(𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠)))
3836, 37sylib 208 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → ∀𝑏(𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠)))
3938ex 449 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑠𝑎 → ∀𝑏(𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠))))
40 sp 2091 . . . . . . . . . . . . . . . . . . 19 (∀𝑏(𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠)) → (𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠)))
4139, 40syl6 35 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑠𝑎 → (𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠))))
4241com23 86 . . . . . . . . . . . . . . . . 17 ((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑏 ∈ (𝐾𝑎) → (𝑠𝑎 → ¬ 𝑏 ∈ (𝐾𝑠))))
43423adant1r 1359 . . . . . . . . . . . . . . . 16 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑏 ∈ (𝐾𝑎) → (𝑠𝑎 → ¬ 𝑏 ∈ (𝐾𝑠))))
4429, 43mpd 15 . . . . . . . . . . . . . . 15 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑠𝑎 → ¬ 𝑏 ∈ (𝐾𝑠)))
4544necon4ad 2842 . . . . . . . . . . . . . 14 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑏 ∈ (𝐾𝑠) → 𝑠 = 𝑎))
46453expia 1286 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → (𝑠 ∈ ω → (𝑏 ∈ (𝐾𝑠) → 𝑠 = 𝑎)))
4746impd 446 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠)) → 𝑠 = 𝑎))
48 eleq1 2718 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑎 → (𝑠 ∈ ω ↔ 𝑎 ∈ ω))
49 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑠 = 𝑎 → (𝐾𝑠) = (𝐾𝑎))
5049eleq2d 2716 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑎 → (𝑏 ∈ (𝐾𝑠) ↔ 𝑏 ∈ (𝐾𝑎)))
5148, 50anbi12d 747 . . . . . . . . . . . . . . 15 (𝑠 = 𝑎 → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠)) ↔ (𝑎 ∈ ω ∧ 𝑏 ∈ (𝐾𝑎))))
5251biimprcd 240 . . . . . . . . . . . . . 14 ((𝑎 ∈ ω ∧ 𝑏 ∈ (𝐾𝑎)) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
5352ancoms 468 . . . . . . . . . . . . 13 ((𝑏 ∈ (𝐾𝑎) ∧ 𝑎 ∈ ω) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
5453adantll 750 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
5547, 54impbid 202 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠)) ↔ 𝑠 = 𝑎))
5655iota5 5909 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))) = 𝑎)
5756an32s 863 . . . . . . . . 9 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))) = 𝑎)
5828, 57eqtr2d 2686 . . . . . . . 8 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → 𝑎 = (𝐿𝑏))
5922, 58jca 553 . . . . . . 7 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → (𝑏𝐺𝑎 = (𝐿𝑏)))
6059ex 449 . . . . . 6 ((𝜑𝑎 ∈ ω) → (𝑏 ∈ (𝐾𝑎) → (𝑏𝐺𝑎 = (𝐿𝑏))))
6160eximdv 1886 . . . . 5 ((𝜑𝑎 ∈ ω) → (∃𝑏 𝑏 ∈ (𝐾𝑎) → ∃𝑏(𝑏𝐺𝑎 = (𝐿𝑏))))
62 df-rex 2947 . . . . 5 (∃𝑏𝐺 𝑎 = (𝐿𝑏) ↔ ∃𝑏(𝑏𝐺𝑎 = (𝐿𝑏)))
6361, 62syl6ibr 242 . . . 4 ((𝜑𝑎 ∈ ω) → (∃𝑏 𝑏 ∈ (𝐾𝑎) → ∃𝑏𝐺 𝑎 = (𝐿𝑏)))
6420, 63mpd 15 . . 3 ((𝜑𝑎 ∈ ω) → ∃𝑏𝐺 𝑎 = (𝐿𝑏))
6564ralrimiva 2995 . 2 (𝜑 → ∀𝑎 ∈ ω ∃𝑏𝐺 𝑎 = (𝐿𝑏))
66 dffo3 6414 . 2 (𝐿:𝐺onto→ω ↔ (𝐿:𝐺⟶ω ∧ ∀𝑎 ∈ ω ∃𝑏𝐺 𝑎 = (𝐿𝑏)))
6711, 65, 66sylanbrc 699 1 (𝜑𝐿:𝐺onto→ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1054  wal 1521   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  {cab 2637  wne 2823  wral 2941  wrex 2942  {crab 2945  cdif 3604  cin 3606  wss 3607  wpss 3608  c0 3948  𝒫 cpw 4191   cint 4507   class class class wbr 4685  cmpt 4762  ran crn 5144  ccom 5147  suc csuc 5763  cio 5887  wf 5922  ontowfo 5924  cfv 5926  crio 6650  ωcom 7107  cen 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-1o 7605  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803
This theorem is referenced by:  isf32lem10  9222
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