MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fin23lem39 Structured version   Visualization version   GIF version

Theorem fin23lem39 10329
Description: Lemma for fin23 10368. 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 10328 . 2 (𝜑 → ¬ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
71, 2, 3, 4, 5fin23lem35 10326 . . . . . . 7 ((𝜑𝑒 ∈ ω) → ran (𝑌‘suc 𝑒) ⊊ ran (𝑌𝑒))
87pssssd 4094 . . . . . 6 ((𝜑𝑒 ∈ ω) → ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒))
9 peano2 7865 . . . . . . . . 9 (𝑒 ∈ ω → suc 𝑒 ∈ ω)
10 fveq2 6879 . . . . . . . . . . . 12 (𝑐 = suc 𝑒 → (𝑌𝑐) = (𝑌‘suc 𝑒))
1110rneqd 5930 . . . . . . . . . . 11 (𝑐 = suc 𝑒 → ran (𝑌𝑐) = ran (𝑌‘suc 𝑒))
1211unieqd 4916 . . . . . . . . . 10 (𝑐 = suc 𝑒 ran (𝑌𝑐) = ran (𝑌‘suc 𝑒))
13 eqid 2732 . . . . . . . . . 10 (𝑐 ∈ ω ↦ ran (𝑌𝑐)) = (𝑐 ∈ ω ↦ ran (𝑌𝑐))
14 fvex 6892 . . . . . . . . . . . 12 (𝑌‘suc 𝑒) ∈ V
1514rnex 7887 . . . . . . . . . . 11 ran (𝑌‘suc 𝑒) ∈ V
1615uniex 7715 . . . . . . . . . 10 ran (𝑌‘suc 𝑒) ∈ V
1712, 13, 16fvmpt 6985 . . . . . . . . 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 6879 . . . . . . . . . . 11 (𝑐 = 𝑒 → (𝑌𝑐) = (𝑌𝑒))
2019rneqd 5930 . . . . . . . . . 10 (𝑐 = 𝑒 → ran (𝑌𝑐) = ran (𝑌𝑒))
2120unieqd 4916 . . . . . . . . 9 (𝑐 = 𝑒 ran (𝑌𝑐) = ran (𝑌𝑒))
22 fvex 6892 . . . . . . . . . . 11 (𝑌𝑒) ∈ V
2322rnex 7887 . . . . . . . . . 10 ran (𝑌𝑒) ∈ V
2423uniex 7715 . . . . . . . . 9 ran (𝑌𝑒) ∈ V
2521, 13, 24fvmpt 6985 . . . . . . . 8 (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) = ran (𝑌𝑒))
2618, 25sseq12d 4012 . . . . . . 7 (𝑒 ∈ ω → (((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) ↔ ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒)))
2726adantl 482 . . . . . 6 ((𝜑𝑒 ∈ ω) → (((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) ↔ ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒)))
288, 27mpbird 256 . . . . 5 ((𝜑𝑒 ∈ ω) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
2928ralrimiva 3146 . . . 4 (𝜑 → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
3029adantr 481 . . 3 ((𝜑𝐺𝐹) → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
31 fveq1 6878 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (𝑑‘suc 𝑒) = ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒))
32 fveq1 6878 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (𝑑𝑒) = ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
3331, 32sseq12d 4012 . . . . . 6 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ((𝑑‘suc 𝑒) ⊆ (𝑑𝑒) ↔ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒)))
3433ralbidv 3177 . . . . 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 5928 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
3635inteqd 4949 . . . . . 6 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
3736, 35eleq12d 2827 . . . . 5 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ( ran 𝑑 ∈ ran 𝑑 ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐))))
3834, 37imbi12d 344 . . . 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 10308 . . . . . 6 (𝐺𝐹 → (𝐺𝐹 ↔ ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑)))
4039ibi 266 . . . . 5 (𝐺𝐹 → ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑))
4140adantl 482 . . . 4 ((𝜑𝐺𝐹) → ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑))
421, 2, 3, 4, 5fin23lem34 10325 . . . . . . . . 9 ((𝜑𝑐 ∈ ω) → ((𝑌𝑐):ω–1-1→V ∧ ran (𝑌𝑐) ⊆ 𝐺))
4342simprd 496 . . . . . . . 8 ((𝜑𝑐 ∈ ω) → ran (𝑌𝑐) ⊆ 𝐺)
4443adantlr 713 . . . . . . 7 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ran (𝑌𝑐) ⊆ 𝐺)
45 elpw2g 5338 . . . . . . . 8 (𝐺𝐹 → ( ran (𝑌𝑐) ∈ 𝒫 𝐺 ran (𝑌𝑐) ⊆ 𝐺))
4645ad2antlr 725 . . . . . . 7 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ( ran (𝑌𝑐) ∈ 𝒫 𝐺 ran (𝑌𝑐) ⊆ 𝐺))
4744, 46mpbird 256 . . . . . 6 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ran (𝑌𝑐) ∈ 𝒫 𝐺)
4847fmpttd 7100 . . . . 5 ((𝜑𝐺𝐹) → (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺)
49 pwexg 5370 . . . . . 6 (𝐺𝐹 → 𝒫 𝐺 ∈ V)
50 vex 3478 . . . . . . . 8 ∈ V
51 f1f 6775 . . . . . . . 8 (:ω–1-1→V → :ω⟶V)
52 dmfex 7882 . . . . . . . 8 (( ∈ V ∧ :ω⟶V) → ω ∈ V)
5350, 51, 52sylancr 587 . . . . . . 7 (:ω–1-1→V → ω ∈ V)
542, 53syl 17 . . . . . 6 (𝜑 → ω ∈ V)
55 elmapg 8818 . . . . . 6 ((𝒫 𝐺 ∈ V ∧ ω ∈ V) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺m ω) ↔ (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺))
5649, 54, 55syl2anr 597 . . . . 5 ((𝜑𝐺𝐹) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺m ω) ↔ (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺))
5748, 56mpbird 256 . . . 4 ((𝜑𝐺𝐹) → (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺m ω))
5838, 41, 57rspcdva 3611 . . 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 814 1 (𝜑 → ¬ 𝐺𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  {cab 2709  wral 3061  Vcvv 3474  wss 3945  wpss 3946  𝒫 cpw 4597   cuni 4902   cint 4944  cmpt 5225  ran crn 5671  cres 5672  suc csuc 6356  wf 6529  1-1wf1 6530  cfv 6533  (class class class)co 7394  ωcom 7839  reccrdg 8393  m cmap 8805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5293  ax-nul 5300  ax-pow 5357  ax-pr 5421  ax-un 7709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5568  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7397  df-oprab 7398  df-mpo 7399  df-om 7840  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8355  df-rdg 8394  df-map 8807
This theorem is referenced by:  fin23lem41  10331
  Copyright terms: Public domain W3C validator