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Theorem fin23lem39 9374
Description: Lemma for fin23 9413. Thus, we have that 𝑔 could not have been in 𝐹 after all. (Contributed by Stefan O'Rear, 4-Nov-2014.)
Hypotheses
Ref Expression
fin23lem33.f 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
fin23lem.f (𝜑:ω–1-1→V)
fin23lem.g (𝜑 ran 𝐺)
fin23lem.h (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
fin23lem.i 𝑌 = (rec(𝑖, ) ↾ ω)
Assertion
Ref Expression
fin23lem39 (𝜑 → ¬ 𝐺𝐹)
Distinct variable groups:   𝑔,𝑎,𝑖,𝑗,𝑥,,𝐺   𝐹,𝑎   𝜑,𝑎,𝑗   𝑌,𝑎,𝑗
Allowed substitution hints:   𝜑(𝑥,𝑔,,𝑖)   𝐹(𝑥,𝑔,,𝑖,𝑗)   𝑌(𝑥,𝑔,,𝑖)

Proof of Theorem fin23lem39
Dummy variables 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin23lem33.f . . 3 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
2 fin23lem.f . . 3 (𝜑:ω–1-1→V)
3 fin23lem.g . . 3 (𝜑 ran 𝐺)
4 fin23lem.h . . 3 (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
5 fin23lem.i . . 3 𝑌 = (rec(𝑖, ) ↾ ω)
61, 2, 3, 4, 5fin23lem38 9373 . 2 (𝜑 → ¬ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
71, 2, 3, 4, 5fin23lem35 9371 . . . . . . 7 ((𝜑𝑒 ∈ ω) → ran (𝑌‘suc 𝑒) ⊊ ran (𝑌𝑒))
87pssssd 3854 . . . . . 6 ((𝜑𝑒 ∈ ω) → ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒))
9 peano2 7233 . . . . . . . . 9 (𝑒 ∈ ω → suc 𝑒 ∈ ω)
10 fveq2 6332 . . . . . . . . . . . 12 (𝑐 = suc 𝑒 → (𝑌𝑐) = (𝑌‘suc 𝑒))
1110rneqd 5491 . . . . . . . . . . 11 (𝑐 = suc 𝑒 → ran (𝑌𝑐) = ran (𝑌‘suc 𝑒))
1211unieqd 4584 . . . . . . . . . 10 (𝑐 = suc 𝑒 ran (𝑌𝑐) = ran (𝑌‘suc 𝑒))
13 eqid 2771 . . . . . . . . . 10 (𝑐 ∈ ω ↦ ran (𝑌𝑐)) = (𝑐 ∈ ω ↦ ran (𝑌𝑐))
14 fvex 6342 . . . . . . . . . . . 12 (𝑌‘suc 𝑒) ∈ V
1514rnex 7247 . . . . . . . . . . 11 ran (𝑌‘suc 𝑒) ∈ V
1615uniex 7100 . . . . . . . . . 10 ran (𝑌‘suc 𝑒) ∈ V
1712, 13, 16fvmpt 6424 . . . . . . . . 9 (suc 𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) = ran (𝑌‘suc 𝑒))
189, 17syl 17 . . . . . . . 8 (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) = ran (𝑌‘suc 𝑒))
19 fveq2 6332 . . . . . . . . . . 11 (𝑐 = 𝑒 → (𝑌𝑐) = (𝑌𝑒))
2019rneqd 5491 . . . . . . . . . 10 (𝑐 = 𝑒 → ran (𝑌𝑐) = ran (𝑌𝑒))
2120unieqd 4584 . . . . . . . . 9 (𝑐 = 𝑒 ran (𝑌𝑐) = ran (𝑌𝑒))
22 fvex 6342 . . . . . . . . . . 11 (𝑌𝑒) ∈ V
2322rnex 7247 . . . . . . . . . 10 ran (𝑌𝑒) ∈ V
2423uniex 7100 . . . . . . . . 9 ran (𝑌𝑒) ∈ V
2521, 13, 24fvmpt 6424 . . . . . . . 8 (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) = ran (𝑌𝑒))
2618, 25sseq12d 3783 . . . . . . 7 (𝑒 ∈ ω → (((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) ↔ ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒)))
2726adantl 467 . . . . . 6 ((𝜑𝑒 ∈ ω) → (((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) ↔ ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒)))
288, 27mpbird 247 . . . . 5 ((𝜑𝑒 ∈ ω) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
2928ralrimiva 3115 . . . 4 (𝜑 → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
3029adantr 466 . . 3 ((𝜑𝐺𝐹) → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
31 fveq1 6331 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (𝑑‘suc 𝑒) = ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒))
32 fveq1 6331 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (𝑑𝑒) = ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
3331, 32sseq12d 3783 . . . . . 6 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ((𝑑‘suc 𝑒) ⊆ (𝑑𝑒) ↔ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒)))
3433ralbidv 3135 . . . . 5 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) ↔ ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒)))
35 rneq 5489 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
3635inteqd 4616 . . . . . 6 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
3736, 35eleq12d 2844 . . . . 5 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ( ran 𝑑 ∈ ran 𝑑 ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐))))
3834, 37imbi12d 333 . . . 4 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ((∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑) ↔ (∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) → ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))))
391isfin3ds 9353 . . . . . 6 (𝐺𝐹 → (𝐺𝐹 ↔ ∀𝑑 ∈ (𝒫 𝐺𝑚 ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑)))
4039ibi 256 . . . . 5 (𝐺𝐹 → ∀𝑑 ∈ (𝒫 𝐺𝑚 ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑))
4140adantl 467 . . . 4 ((𝜑𝐺𝐹) → ∀𝑑 ∈ (𝒫 𝐺𝑚 ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑))
421, 2, 3, 4, 5fin23lem34 9370 . . . . . . . . 9 ((𝜑𝑐 ∈ ω) → ((𝑌𝑐):ω–1-1→V ∧ ran (𝑌𝑐) ⊆ 𝐺))
4342simprd 477 . . . . . . . 8 ((𝜑𝑐 ∈ ω) → ran (𝑌𝑐) ⊆ 𝐺)
4443adantlr 686 . . . . . . 7 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ran (𝑌𝑐) ⊆ 𝐺)
45 elpw2g 4958 . . . . . . . 8 (𝐺𝐹 → ( ran (𝑌𝑐) ∈ 𝒫 𝐺 ran (𝑌𝑐) ⊆ 𝐺))
4645ad2antlr 698 . . . . . . 7 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ( ran (𝑌𝑐) ∈ 𝒫 𝐺 ran (𝑌𝑐) ⊆ 𝐺))
4744, 46mpbird 247 . . . . . 6 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ran (𝑌𝑐) ∈ 𝒫 𝐺)
4847, 13fmptd 6527 . . . . 5 ((𝜑𝐺𝐹) → (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺)
49 pwexg 4981 . . . . . 6 (𝐺𝐹 → 𝒫 𝐺 ∈ V)
50 vex 3354 . . . . . . . 8 ∈ V
51 f1f 6241 . . . . . . . 8 (:ω–1-1→V → :ω⟶V)
52 dmfex 7271 . . . . . . . 8 (( ∈ V ∧ :ω⟶V) → ω ∈ V)
5350, 51, 52sylancr 567 . . . . . . 7 (:ω–1-1→V → ω ∈ V)
542, 53syl 17 . . . . . 6 (𝜑 → ω ∈ V)
55 elmapg 8022 . . . . . 6 ((𝒫 𝐺 ∈ V ∧ ω ∈ V) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺𝑚 ω) ↔ (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺))
5649, 54, 55syl2anr 576 . . . . 5 ((𝜑𝐺𝐹) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺𝑚 ω) ↔ (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺))
5748, 56mpbird 247 . . . 4 ((𝜑𝐺𝐹) → (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺𝑚 ω))
5838, 41, 57rspcdva 3466 . . 3 ((𝜑𝐺𝐹) → (∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) → ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐))))
5930, 58mpd 15 . 2 ((𝜑𝐺𝐹) → ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
606, 59mtand 799 1 (𝜑 → ¬ 𝐺𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wal 1629   = wceq 1631  wcel 2145  {cab 2757  wral 3061  Vcvv 3351  wss 3723  wpss 3724  𝒫 cpw 4297   cuni 4574   cint 4611  cmpt 4863  ran crn 5250  cres 5251  suc csuc 5868  wf 6027  1-1wf1 6028  cfv 6031  (class class class)co 6793  ωcom 7212  reccrdg 7658  𝑚 cmap 8009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  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-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-map 8011
This theorem is referenced by:  fin23lem41  9376
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