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Theorem isf32lem9 9384
Description: Lemma for isfin3-2 9390. 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 3833 . . . . . . 7 {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))} ⊆ ω
3 iotacl 6017 . . . . . . 7 (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))})
42, 3sseldi 3748 . . . . . 6 (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ ω)
5 iotanul 6009 . . . . . . 7 (¬ ∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) = ∅)
6 peano1 7231 . . . . . . 7 ∅ ∈ ω
75, 6syl6eqel 2857 . . . . . 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 6525 . . 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 9381 . . . . 5 ((𝜑𝑎 ∈ ω) → (𝐾𝑎) ≠ ∅)
19 n0 4076 . . . . 5 ((𝐾𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐾𝑎))
2018, 19sylib 208 . . . 4 ((𝜑𝑎 ∈ ω) → ∃𝑏 𝑏 ∈ (𝐾𝑎))
2112, 13, 14, 15, 16, 17isf32lem8 9383 . . . . . . . . 9 ((𝜑𝑎 ∈ ω) → (𝐾𝑎) ⊆ 𝐺)
2221sselda 3750 . . . . . . . 8 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → 𝑏𝐺)
23 eleq1w 2832 . . . . . . . . . . . . 13 (𝑡 = 𝑏 → (𝑡 ∈ (𝐾𝑠) ↔ 𝑏 ∈ (𝐾𝑠)))
2423anbi2d 606 . . . . . . . . . . . 12 (𝑡 = 𝑏 → ((𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) ↔ (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
2524iotabidv 6015 . . . . . . . . . . 11 (𝑡 = 𝑏 → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
26 iotaex 6011 . . . . . . . . . . 11 (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ V
2725, 1, 26fvmpt3i 6429 . . . . . . . . . 10 (𝑏𝐺 → (𝐿𝑏) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
2822, 27syl 17 . . . . . . . . 9 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → (𝐿𝑏) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
29 simp1r 1239 . . . . . . . . . . . . . . . 16 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → 𝑏 ∈ (𝐾𝑎))
30 simpl1 1226 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝜑)
31 simpr 471 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑠𝑎)
3231necomd 2997 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑎𝑠)
33 simpl2 1228 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑎 ∈ ω)
34 simpl3 1230 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑠 ∈ ω)
3512, 13, 14, 15, 16, 17isf32lem7 9382 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎𝑠) ∧ (𝑎 ∈ ω ∧ 𝑠 ∈ ω)) → ((𝐾𝑎) ∩ (𝐾𝑠)) = ∅)
3630, 32, 33, 34, 35syl22anc 1476 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → ((𝐾𝑎) ∩ (𝐾𝑠)) = ∅)
37 disj1 4160 . . . . . . . . . . . . . . . . . . . . 21 (((𝐾𝑎) ∩ (𝐾𝑠)) = ∅ ↔ ∀𝑏(𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠)))
3836, 37sylib 208 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → ∀𝑏(𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠)))
3938ex 397 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑠𝑎 → ∀𝑏(𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠))))
40 sp 2206 . . . . . . . . . . . . . . . . . . 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 1186 . . . . . . . . . . . . . . . 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 2961 . . . . . . . . . . . . . 14 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑏 ∈ (𝐾𝑠) → 𝑠 = 𝑎))
46453expia 1113 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → (𝑠 ∈ ω → (𝑏 ∈ (𝐾𝑠) → 𝑠 = 𝑎)))
4746impd 396 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠)) → 𝑠 = 𝑎))
48 eleq1w 2832 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑎 → (𝑠 ∈ ω ↔ 𝑎 ∈ ω))
49 fveq2 6332 . . . . . . . . . . . . . . . . 17 (𝑠 = 𝑎 → (𝐾𝑠) = (𝐾𝑎))
5049eleq2d 2835 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑎 → (𝑏 ∈ (𝐾𝑠) ↔ 𝑏 ∈ (𝐾𝑎)))
5148, 50anbi12d 608 . . . . . . . . . . . . . . 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 455 . . . . . . . . . . . . 13 ((𝑏 ∈ (𝐾𝑎) ∧ 𝑎 ∈ ω) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
5453adantll 685 . . . . . . . . . . . 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 6014 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))) = 𝑎)
5756an32s 623 . . . . . . . . 9 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))) = 𝑎)
5828, 57eqtr2d 2805 . . . . . . . 8 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → 𝑎 = (𝐿𝑏))
5922, 58jca 495 . . . . . . 7 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → (𝑏𝐺𝑎 = (𝐿𝑏)))
6059ex 397 . . . . . 6 ((𝜑𝑎 ∈ ω) → (𝑏 ∈ (𝐾𝑎) → (𝑏𝐺𝑎 = (𝐿𝑏))))
6160eximdv 1997 . . . . 5 ((𝜑𝑎 ∈ ω) → (∃𝑏 𝑏 ∈ (𝐾𝑎) → ∃𝑏(𝑏𝐺𝑎 = (𝐿𝑏))))
62 df-rex 3066 . . . . 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 3114 . 2 (𝜑 → ∀𝑎 ∈ ω ∃𝑏𝐺 𝑎 = (𝐿𝑏))
66 dffo3 6517 . 2 (𝐿:𝐺onto→ω ↔ (𝐿:𝐺⟶ω ∧ ∀𝑎 ∈ ω ∃𝑏𝐺 𝑎 = (𝐿𝑏)))
6711, 65, 66sylanbrc 564 1 (𝜑𝐿:𝐺onto→ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1070  wal 1628   = wceq 1630  wex 1851  wcel 2144  ∃!weu 2617  {cab 2756  wne 2942  wral 3060  wrex 3061  {crab 3064  cdif 3718  cin 3720  wss 3721  wpss 3722  c0 4061  𝒫 cpw 4295   cint 4609   class class class wbr 4784  cmpt 4861  ran crn 5250  ccom 5253  suc csuc 5868  cio 5992  wf 6027  ontowfo 6029  cfv 6031  crio 6752  ωcom 7211  cen 8105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6753  df-om 7212  df-wrecs 7558  df-recs 7620  df-1o 7712  df-er 7895  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964
This theorem is referenced by:  isf32lem10  9385
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