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Theorem fin23lem39 9991
Description: Lemma for fin23 10030. 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 9990 . 2 (𝜑 → ¬ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
71, 2, 3, 4, 5fin23lem35 9988 . . . . . . 7 ((𝜑𝑒 ∈ ω) → ran (𝑌‘suc 𝑒) ⊊ ran (𝑌𝑒))
87pssssd 4028 . . . . . 6 ((𝜑𝑒 ∈ ω) → ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒))
9 peano2 7689 . . . . . . . . 9 (𝑒 ∈ ω → suc 𝑒 ∈ ω)
10 fveq2 6738 . . . . . . . . . . . 12 (𝑐 = suc 𝑒 → (𝑌𝑐) = (𝑌‘suc 𝑒))
1110rneqd 5824 . . . . . . . . . . 11 (𝑐 = suc 𝑒 → ran (𝑌𝑐) = ran (𝑌‘suc 𝑒))
1211unieqd 4849 . . . . . . . . . 10 (𝑐 = suc 𝑒 ran (𝑌𝑐) = ran (𝑌‘suc 𝑒))
13 eqid 2739 . . . . . . . . . 10 (𝑐 ∈ ω ↦ ran (𝑌𝑐)) = (𝑐 ∈ ω ↦ ran (𝑌𝑐))
14 fvex 6751 . . . . . . . . . . . 12 (𝑌‘suc 𝑒) ∈ V
1514rnex 7711 . . . . . . . . . . 11 ran (𝑌‘suc 𝑒) ∈ V
1615uniex 7550 . . . . . . . . . 10 ran (𝑌‘suc 𝑒) ∈ V
1712, 13, 16fvmpt 6839 . . . . . . . . 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 6738 . . . . . . . . . . 11 (𝑐 = 𝑒 → (𝑌𝑐) = (𝑌𝑒))
2019rneqd 5824 . . . . . . . . . 10 (𝑐 = 𝑒 → ran (𝑌𝑐) = ran (𝑌𝑒))
2120unieqd 4849 . . . . . . . . 9 (𝑐 = 𝑒 ran (𝑌𝑐) = ran (𝑌𝑒))
22 fvex 6751 . . . . . . . . . . 11 (𝑌𝑒) ∈ V
2322rnex 7711 . . . . . . . . . 10 ran (𝑌𝑒) ∈ V
2423uniex 7550 . . . . . . . . 9 ran (𝑌𝑒) ∈ V
2521, 13, 24fvmpt 6839 . . . . . . . 8 (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) = ran (𝑌𝑒))
2618, 25sseq12d 3950 . . . . . . 7 (𝑒 ∈ ω → (((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) ↔ ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒)))
2726adantl 485 . . . . . 6 ((𝜑𝑒 ∈ ω) → (((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) ↔ ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒)))
288, 27mpbird 260 . . . . 5 ((𝜑𝑒 ∈ ω) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
2928ralrimiva 3107 . . . 4 (𝜑 → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
3029adantr 484 . . 3 ((𝜑𝐺𝐹) → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
31 fveq1 6737 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (𝑑‘suc 𝑒) = ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒))
32 fveq1 6737 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (𝑑𝑒) = ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
3331, 32sseq12d 3950 . . . . . 6 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ((𝑑‘suc 𝑒) ⊆ (𝑑𝑒) ↔ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒)))
3433ralbidv 3120 . . . . 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 5822 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
3635inteqd 4880 . . . . . 6 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
3736, 35eleq12d 2834 . . . . 5 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ( ran 𝑑 ∈ ran 𝑑 ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐))))
3834, 37imbi12d 348 . . . 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 9970 . . . . . 6 (𝐺𝐹 → (𝐺𝐹 ↔ ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑)))
4039ibi 270 . . . . 5 (𝐺𝐹 → ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑))
4140adantl 485 . . . 4 ((𝜑𝐺𝐹) → ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑))
421, 2, 3, 4, 5fin23lem34 9987 . . . . . . . . 9 ((𝜑𝑐 ∈ ω) → ((𝑌𝑐):ω–1-1→V ∧ ran (𝑌𝑐) ⊆ 𝐺))
4342simprd 499 . . . . . . . 8 ((𝜑𝑐 ∈ ω) → ran (𝑌𝑐) ⊆ 𝐺)
4443adantlr 715 . . . . . . 7 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ran (𝑌𝑐) ⊆ 𝐺)
45 elpw2g 5253 . . . . . . . 8 (𝐺𝐹 → ( ran (𝑌𝑐) ∈ 𝒫 𝐺 ran (𝑌𝑐) ⊆ 𝐺))
4645ad2antlr 727 . . . . . . 7 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ( ran (𝑌𝑐) ∈ 𝒫 𝐺 ran (𝑌𝑐) ⊆ 𝐺))
4744, 46mpbird 260 . . . . . 6 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ran (𝑌𝑐) ∈ 𝒫 𝐺)
4847fmpttd 6953 . . . . 5 ((𝜑𝐺𝐹) → (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺)
49 pwexg 5287 . . . . . 6 (𝐺𝐹 → 𝒫 𝐺 ∈ V)
50 vex 3426 . . . . . . . 8 ∈ V
51 f1f 6636 . . . . . . . 8 (:ω–1-1→V → :ω⟶V)
52 dmfex 7706 . . . . . . . 8 (( ∈ V ∧ :ω⟶V) → ω ∈ V)
5350, 51, 52sylancr 590 . . . . . . 7 (:ω–1-1→V → ω ∈ V)
542, 53syl 17 . . . . . 6 (𝜑 → ω ∈ V)
55 elmapg 8544 . . . . . 6 ((𝒫 𝐺 ∈ V ∧ ω ∈ V) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺m ω) ↔ (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺))
5649, 54, 55syl2anr 600 . . . . 5 ((𝜑𝐺𝐹) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺m ω) ↔ (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺))
5748, 56mpbird 260 . . . 4 ((𝜑𝐺𝐹) → (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺m ω))
5838, 41, 57rspcdva 3553 . . 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 816 1 (𝜑 → ¬ 𝐺𝐹)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1541   = wceq 1543  wcel 2112  {cab 2716  wral 3063  Vcvv 3422  wss 3882  wpss 3883  𝒫 cpw 4529   cuni 4835   cint 4875  cmpt 5151  ran crn 5569  cres 5570  suc csuc 6235  wf 6396  1-1wf1 6397  cfv 6400  (class class class)co 7234  ωcom 7665  reccrdg 8168  m cmap 8531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3711  df-csb 3828  df-dif 3885  df-un 3887  df-in 3889  df-ss 3899  df-pss 3901  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4836  df-int 4876  df-iun 4922  df-br 5070  df-opab 5132  df-mpt 5152  df-tr 5178  df-id 5471  df-eprel 5477  df-po 5485  df-so 5486  df-fr 5526  df-we 5528  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ima 5581  df-pred 6178  df-ord 6236  df-on 6237  df-lim 6238  df-suc 6239  df-iota 6358  df-fun 6402  df-fn 6403  df-f 6404  df-f1 6405  df-fo 6406  df-f1o 6407  df-fv 6408  df-ov 7237  df-oprab 7238  df-mpo 7239  df-om 7666  df-wrecs 8070  df-recs 8131  df-rdg 8169  df-map 8533
This theorem is referenced by:  fin23lem41  9993
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