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Theorem fin23lem39 10388
Description: Lemma for fin23 10427. 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 10387 . 2 (𝜑 → ¬ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
71, 2, 3, 4, 5fin23lem35 10385 . . . . . . 7 ((𝜑𝑒 ∈ ω) → ran (𝑌‘suc 𝑒) ⊊ ran (𝑌𝑒))
87pssssd 4110 . . . . . 6 ((𝜑𝑒 ∈ ω) → ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒))
9 peano2 7913 . . . . . . . . 9 (𝑒 ∈ ω → suc 𝑒 ∈ ω)
10 fveq2 6907 . . . . . . . . . . . 12 (𝑐 = suc 𝑒 → (𝑌𝑐) = (𝑌‘suc 𝑒))
1110rneqd 5952 . . . . . . . . . . 11 (𝑐 = suc 𝑒 → ran (𝑌𝑐) = ran (𝑌‘suc 𝑒))
1211unieqd 4925 . . . . . . . . . 10 (𝑐 = suc 𝑒 ran (𝑌𝑐) = ran (𝑌‘suc 𝑒))
13 eqid 2735 . . . . . . . . . 10 (𝑐 ∈ ω ↦ ran (𝑌𝑐)) = (𝑐 ∈ ω ↦ ran (𝑌𝑐))
14 fvex 6920 . . . . . . . . . . . 12 (𝑌‘suc 𝑒) ∈ V
1514rnex 7933 . . . . . . . . . . 11 ran (𝑌‘suc 𝑒) ∈ V
1615uniex 7760 . . . . . . . . . 10 ran (𝑌‘suc 𝑒) ∈ V
1712, 13, 16fvmpt 7016 . . . . . . . . 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 6907 . . . . . . . . . . 11 (𝑐 = 𝑒 → (𝑌𝑐) = (𝑌𝑒))
2019rneqd 5952 . . . . . . . . . 10 (𝑐 = 𝑒 → ran (𝑌𝑐) = ran (𝑌𝑒))
2120unieqd 4925 . . . . . . . . 9 (𝑐 = 𝑒 ran (𝑌𝑐) = ran (𝑌𝑒))
22 fvex 6920 . . . . . . . . . . 11 (𝑌𝑒) ∈ V
2322rnex 7933 . . . . . . . . . 10 ran (𝑌𝑒) ∈ V
2423uniex 7760 . . . . . . . . 9 ran (𝑌𝑒) ∈ V
2521, 13, 24fvmpt 7016 . . . . . . . 8 (𝑒 ∈ ω → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) = ran (𝑌𝑒))
2618, 25sseq12d 4029 . . . . . . 7 (𝑒 ∈ ω → (((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) ↔ ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒)))
2726adantl 481 . . . . . 6 ((𝜑𝑒 ∈ ω) → (((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒) ↔ ran (𝑌‘suc 𝑒) ⊆ ran (𝑌𝑒)))
288, 27mpbird 257 . . . . 5 ((𝜑𝑒 ∈ ω) → ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
2928ralrimiva 3144 . . . 4 (𝜑 → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
3029adantr 480 . . 3 ((𝜑𝐺𝐹) → ∀𝑒 ∈ ω ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
31 fveq1 6906 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (𝑑‘suc 𝑒) = ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒))
32 fveq1 6906 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → (𝑑𝑒) = ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒))
3331, 32sseq12d 4029 . . . . . 6 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ((𝑑‘suc 𝑒) ⊆ (𝑑𝑒) ↔ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘suc 𝑒) ⊆ ((𝑐 ∈ ω ↦ ran (𝑌𝑐))‘𝑒)))
3433ralbidv 3176 . . . . 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 5950 . . . . . . 7 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
3635inteqd 4956 . . . . . 6 (𝑑 = (𝑐 ∈ ω ↦ ran (𝑌𝑐)) → ran 𝑑 = ran (𝑐 ∈ ω ↦ ran (𝑌𝑐)))
3736, 35eleq12d 2833 . . . . 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 10367 . . . . . 6 (𝐺𝐹 → (𝐺𝐹 ↔ ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑)))
4039ibi 267 . . . . 5 (𝐺𝐹 → ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑))
4140adantl 481 . . . 4 ((𝜑𝐺𝐹) → ∀𝑑 ∈ (𝒫 𝐺m ω)(∀𝑒 ∈ ω (𝑑‘suc 𝑒) ⊆ (𝑑𝑒) → ran 𝑑 ∈ ran 𝑑))
421, 2, 3, 4, 5fin23lem34 10384 . . . . . . . . 9 ((𝜑𝑐 ∈ ω) → ((𝑌𝑐):ω–1-1→V ∧ ran (𝑌𝑐) ⊆ 𝐺))
4342simprd 495 . . . . . . . 8 ((𝜑𝑐 ∈ ω) → ran (𝑌𝑐) ⊆ 𝐺)
4443adantlr 715 . . . . . . 7 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ran (𝑌𝑐) ⊆ 𝐺)
45 elpw2g 5339 . . . . . . . 8 (𝐺𝐹 → ( ran (𝑌𝑐) ∈ 𝒫 𝐺 ran (𝑌𝑐) ⊆ 𝐺))
4645ad2antlr 727 . . . . . . 7 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ( ran (𝑌𝑐) ∈ 𝒫 𝐺 ran (𝑌𝑐) ⊆ 𝐺))
4744, 46mpbird 257 . . . . . 6 (((𝜑𝐺𝐹) ∧ 𝑐 ∈ ω) → ran (𝑌𝑐) ∈ 𝒫 𝐺)
4847fmpttd 7135 . . . . 5 ((𝜑𝐺𝐹) → (𝑐 ∈ ω ↦ ran (𝑌𝑐)):ω⟶𝒫 𝐺)
49 pwexg 5384 . . . . . 6 (𝐺𝐹 → 𝒫 𝐺 ∈ V)
50 vex 3482 . . . . . . . 8 ∈ V
51 f1f 6805 . . . . . . . 8 (:ω–1-1→V → :ω⟶V)
52 dmfex 7928 . . . . . . . 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 8878 . . . . . 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 257 . . . 4 ((𝜑𝐺𝐹) → (𝑐 ∈ ω ↦ ran (𝑌𝑐)) ∈ (𝒫 𝐺m ω))
5838, 41, 57rspcdva 3623 . . 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 206  wa 395  wal 1535   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  wss 3963  wpss 3964  𝒫 cpw 4605   cuni 4912   cint 4951  cmpt 5231  ran crn 5690  cres 5691  suc csuc 6388  wf 6559  1-1wf1 6560  cfv 6563  (class class class)co 7431  ωcom 7887  reccrdg 8448  m cmap 8865
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-map 8867
This theorem is referenced by:  fin23lem41  10390
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