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Theorem fin23lem39 9775
Description: Lemma for fin23 9814. 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 9774 . 2 (𝜑 → ¬ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
71, 2, 3, 4, 5fin23lem35 9772 . . . . . . 7 ((𝜑𝑒 ∈ ω) → ran (𝑌‘suc 𝑒) ⊊ ran (𝑌𝑒))
87pssssd 4077 . . . . . 6 ((𝜑𝑒 ∈ ω) → ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒))
9 peano2 7605 . . . . . . . . 9 (𝑒 ∈ ω → suc 𝑒 ∈ ω)
10 fveq2 6673 . . . . . . . . . . . 12 (𝑐 = suc 𝑒 → (𝑌𝑐) = (𝑌‘suc 𝑒))
1110rneqd 5811 . . . . . . . . . . 11 (𝑐 = suc 𝑒 → ran (𝑌𝑐) = ran (𝑌‘suc 𝑒))
1211unieqd 4855 . . . . . . . . . 10 (𝑐 = suc 𝑒 ran (𝑌𝑐) = ran (𝑌‘suc 𝑒))
13 eqid 2824 . . . . . . . . . 10 (𝑐 ∈ ω ↦ ran (𝑌𝑐)) = (𝑐 ∈ ω ↦ ran (𝑌𝑐))
14 fvex 6686 . . . . . . . . . . . 12 (𝑌‘suc 𝑒) ∈ V
1514rnex 7620 . . . . . . . . . . 11 ran (𝑌‘suc 𝑒) ∈ V
1615uniex 7470 . . . . . . . . . 10 ran (𝑌‘suc 𝑒) ∈ V
1712, 13, 16fvmpt 6771 . . . . . . . . 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 6673 . . . . . . . . . . 11 (𝑐 = 𝑒 → (𝑌𝑐) = (𝑌𝑒))
2019rneqd 5811 . . . . . . . . . 10 (𝑐 = 𝑒 → ran (𝑌𝑐) = ran (𝑌𝑒))
2120unieqd 4855 . . . . . . . . 9 (𝑐 = 𝑒 ran (𝑌𝑐) = ran (𝑌𝑒))
22 fvex 6686 . . . . . . . . . . 11 (𝑌𝑒) ∈ V
2322rnex 7620 . . . . . . . . . 10 ran (𝑌𝑒) ∈ V
2423uniex 7470 . . . . . . . . 9 ran (𝑌𝑒) ∈ V
2521, 13, 24fvmpt 6771 . . . . . . . 8 (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) = ran (𝑌𝑒))
2618, 25sseq12d 4003 . . . . . . 7 (𝑒 ∈ ω → (((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) ↔ ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒)))
2726adantl 484 . . . . . 6 ((𝜑𝑒 ∈ ω) → (((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) ↔ ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒)))
288, 27mpbird 259 . . . . 5 ((𝜑𝑒 ∈ ω) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
2928ralrimiva 3185 . . . 4 (𝜑 → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
3029adantr 483 . . 3 ((𝜑𝐺𝐹) → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
31 fveq1 6672 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (𝑑‘suc 𝑒) = ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒))
32 fveq1 6672 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (𝑑𝑒) = ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
3331, 32sseq12d 4003 . . . . . 6 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ((𝑑‘suc 𝑒) ⊆ (𝑑𝑒) ↔ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒)))
3433ralbidv 3200 . . . . 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 5809 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
3635inteqd 4884 . . . . . 6 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
3736, 35eleq12d 2910 . . . . 5 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ( ran 𝑑 ∈ ran 𝑑 ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐))))
3834, 37imbi12d 347 . . . 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 9754 . . . . . 6 (𝐺𝐹 → (𝐺𝐹 ↔ ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑)))
4039ibi 269 . . . . 5 (𝐺𝐹 → ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑))
4140adantl 484 . . . 4 ((𝜑𝐺𝐹) → ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑))
421, 2, 3, 4, 5fin23lem34 9771 . . . . . . . . 9 ((𝜑𝑐 ∈ ω) → ((𝑌𝑐):ω–1-1→V ∧ ran (𝑌𝑐) ⊆ 𝐺))
4342simprd 498 . . . . . . . 8 ((𝜑𝑐 ∈ ω) → ran (𝑌𝑐) ⊆ 𝐺)
4443adantlr 713 . . . . . . 7 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ran (𝑌𝑐) ⊆ 𝐺)
45 elpw2g 5250 . . . . . . . 8 (𝐺𝐹 → ( ran (𝑌𝑐) ∈ 𝒫 𝐺 ran (𝑌𝑐) ⊆ 𝐺))
4645ad2antlr 725 . . . . . . 7 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ( ran (𝑌𝑐) ∈ 𝒫 𝐺 ran (𝑌𝑐) ⊆ 𝐺))
4744, 46mpbird 259 . . . . . 6 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ran (𝑌𝑐) ∈ 𝒫 𝐺)
4847fmpttd 6882 . . . . 5 ((𝜑𝐺𝐹) → (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺)
49 pwexg 5282 . . . . . 6 (𝐺𝐹 → 𝒫 𝐺 ∈ V)
50 vex 3500 . . . . . . . 8 ∈ V
51 f1f 6578 . . . . . . . 8 (:ω–1-1→V → :ω⟶V)
52 dmfex 7644 . . . . . . . 8 (( ∈ V ∧ :ω⟶V) → ω ∈ V)
5350, 51, 52sylancr 589 . . . . . . 7 (:ω–1-1→V → ω ∈ V)
542, 53syl 17 . . . . . 6 (𝜑 → ω ∈ V)
55 elmapg 8422 . . . . . 6 ((𝒫 𝐺 ∈ V ∧ ω ∈ V) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺m ω) ↔ (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺))
5649, 54, 55syl2anr 598 . . . . 5 ((𝜑𝐺𝐹) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺m ω) ↔ (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺))
5748, 56mpbird 259 . . . 4 ((𝜑𝐺𝐹) → (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺m ω))
5838, 41, 57rspcdva 3628 . . 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 208  wa 398  wal 1534   = wceq 1536  wcel 2113  {cab 2802  wral 3141  Vcvv 3497  wss 3939  wpss 3940  𝒫 cpw 4542   cuni 4841   cint 4879  cmpt 5149  ran crn 5559  cres 5560  suc csuc 6196  wf 6354  1-1wf1 6355  cfv 6358  (class class class)co 7159  ωcom 7583  reccrdg 8048  m cmap 8409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-map 8411
This theorem is referenced by:  fin23lem41  9777
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