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Theorem isf32lem9 9436
Description: Lemma for isfin3-2 9442. 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 3846 . . . . . . 7 {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))} ⊆ ω
3 iotacl 6054 . . . . . . 7 (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ {𝑠 ∣ (𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))})
42, 3sseldi 3759 . . . . . 6 (∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ ω)
5 iotanul 6046 . . . . . . 7 (¬ ∃!𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) = ∅)
6 peano1 7283 . . . . . . 7 ∅ ∈ ω
75, 6syl6eqel 2852 . . . . . 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 6572 . . 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 9433 . . . . 5 ((𝜑𝑎 ∈ ω) → (𝐾𝑎) ≠ ∅)
19 n0 4095 . . . . 5 ((𝐾𝑎) ≠ ∅ ↔ ∃𝑏 𝑏 ∈ (𝐾𝑎))
2018, 19sylib 209 . . . 4 ((𝜑𝑎 ∈ ω) → ∃𝑏 𝑏 ∈ (𝐾𝑎))
2112, 13, 14, 15, 16, 17isf32lem8 9435 . . . . . . . . 9 ((𝜑𝑎 ∈ ω) → (𝐾𝑎) ⊆ 𝐺)
2221sselda 3761 . . . . . . . 8 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → 𝑏𝐺)
23 eleq1w 2827 . . . . . . . . . . . . 13 (𝑡 = 𝑏 → (𝑡 ∈ (𝐾𝑠) ↔ 𝑏 ∈ (𝐾𝑠)))
2423anbi2d 622 . . . . . . . . . . . 12 (𝑡 = 𝑏 → ((𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠)) ↔ (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
2524iotabidv 6052 . . . . . . . . . . 11 (𝑡 = 𝑏 → (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
26 iotaex 6048 . . . . . . . . . . 11 (℩𝑠(𝑠 ∈ ω ∧ 𝑡 ∈ (𝐾𝑠))) ∈ V
2725, 1, 26fvmpt3i 6476 . . . . . . . . . 10 (𝑏𝐺 → (𝐿𝑏) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
2822, 27syl 17 . . . . . . . . 9 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → (𝐿𝑏) = (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
29 simp1r 1255 . . . . . . . . . . . . . . . 16 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → 𝑏 ∈ (𝐾𝑎))
30 simpl1 1242 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝜑)
31 simpr 477 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑠𝑎)
3231necomd 2992 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑎𝑠)
33 simpl2 1244 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑎 ∈ ω)
34 simpl3 1246 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → 𝑠 ∈ ω)
3512, 13, 14, 15, 16, 17isf32lem7 9434 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑎𝑠) ∧ (𝑎 ∈ ω ∧ 𝑠 ∈ ω)) → ((𝐾𝑎) ∩ (𝐾𝑠)) = ∅)
3630, 32, 33, 34, 35syl22anc 867 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → ((𝐾𝑎) ∩ (𝐾𝑠)) = ∅)
37 disj1 4180 . . . . . . . . . . . . . . . . . . . . 21 (((𝐾𝑎) ∩ (𝐾𝑠)) = ∅ ↔ ∀𝑏(𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠)))
3836, 37sylib 209 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) ∧ 𝑠𝑎) → ∀𝑏(𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠)))
3938ex 401 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑠𝑎 → ∀𝑏(𝑏 ∈ (𝐾𝑎) → ¬ 𝑏 ∈ (𝐾𝑠))))
40 sp 2215 . . . . . . . . . . . . . . . . . . 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 1223 . . . . . . . . . . . . . . . 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 2956 . . . . . . . . . . . . . 14 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω ∧ 𝑠 ∈ ω) → (𝑏 ∈ (𝐾𝑠) → 𝑠 = 𝑎))
46453expia 1150 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → (𝑠 ∈ ω → (𝑏 ∈ (𝐾𝑠) → 𝑠 = 𝑎)))
4746impd 398 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠)) → 𝑠 = 𝑎))
48 eleq1w 2827 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑎 → (𝑠 ∈ ω ↔ 𝑎 ∈ ω))
49 fveq2 6375 . . . . . . . . . . . . . . . . 17 (𝑠 = 𝑎 → (𝐾𝑠) = (𝐾𝑎))
5049eleq2d 2830 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑎 → (𝑏 ∈ (𝐾𝑠) ↔ 𝑏 ∈ (𝐾𝑎)))
5148, 50anbi12d 624 . . . . . . . . . . . . . . 15 (𝑠 = 𝑎 → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠)) ↔ (𝑎 ∈ ω ∧ 𝑏 ∈ (𝐾𝑎))))
5251biimprcd 241 . . . . . . . . . . . . . 14 ((𝑎 ∈ ω ∧ 𝑏 ∈ (𝐾𝑎)) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
5352ancoms 450 . . . . . . . . . . . . 13 ((𝑏 ∈ (𝐾𝑎) ∧ 𝑎 ∈ ω) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
5453adantll 705 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → (𝑠 = 𝑎 → (𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))))
5547, 54impbid 203 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → ((𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠)) ↔ 𝑠 = 𝑎))
5655iota5 6051 . . . . . . . . . 10 (((𝜑𝑏 ∈ (𝐾𝑎)) ∧ 𝑎 ∈ ω) → (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))) = 𝑎)
5756an32s 642 . . . . . . . . 9 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → (℩𝑠(𝑠 ∈ ω ∧ 𝑏 ∈ (𝐾𝑠))) = 𝑎)
5828, 57eqtr2d 2800 . . . . . . . 8 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → 𝑎 = (𝐿𝑏))
5922, 58jca 507 . . . . . . 7 (((𝜑𝑎 ∈ ω) ∧ 𝑏 ∈ (𝐾𝑎)) → (𝑏𝐺𝑎 = (𝐿𝑏)))
6059ex 401 . . . . . 6 ((𝜑𝑎 ∈ ω) → (𝑏 ∈ (𝐾𝑎) → (𝑏𝐺𝑎 = (𝐿𝑏))))
6160eximdv 2012 . . . . 5 ((𝜑𝑎 ∈ ω) → (∃𝑏 𝑏 ∈ (𝐾𝑎) → ∃𝑏(𝑏𝐺𝑎 = (𝐿𝑏))))
62 df-rex 3061 . . . . 5 (∃𝑏𝐺 𝑎 = (𝐿𝑏) ↔ ∃𝑏(𝑏𝐺𝑎 = (𝐿𝑏)))
6361, 62syl6ibr 243 . . . 4 ((𝜑𝑎 ∈ ω) → (∃𝑏 𝑏 ∈ (𝐾𝑎) → ∃𝑏𝐺 𝑎 = (𝐿𝑏)))
6420, 63mpd 15 . . 3 ((𝜑𝑎 ∈ ω) → ∃𝑏𝐺 𝑎 = (𝐿𝑏))
6564ralrimiva 3113 . 2 (𝜑 → ∀𝑎 ∈ ω ∃𝑏𝐺 𝑎 = (𝐿𝑏))
66 dffo3 6564 . 2 (𝐿:𝐺onto→ω ↔ (𝐿:𝐺⟶ω ∧ ∀𝑎 ∈ ω ∃𝑏𝐺 𝑎 = (𝐿𝑏)))
6711, 65, 66sylanbrc 578 1 (𝜑𝐿:𝐺onto→ω)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  ∃!weu 2581  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  cdif 3729  cin 3731  wss 3732  wpss 3733  c0 4079  𝒫 cpw 4315   cint 4633   class class class wbr 4809  cmpt 4888  ran crn 5278  ccom 5281  suc csuc 5910  cio 6029  wf 6064  ontowfo 6066  cfv 6068  crio 6802  ωcom 7263  cen 8157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-om 7264  df-wrecs 7610  df-recs 7672  df-1o 7764  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016
This theorem is referenced by:  isf32lem10  9437
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