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Theorem fin23lem38 9209
Description: Lemma for fin23 9249. The contradictory chain has no minimum. (Contributed by Stefan O'Rear, 2-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
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
fin23lem38 (𝜑 → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
Distinct variable groups:   𝑎,𝑏,𝑔,𝑖,𝑗,𝑥,,𝐺   𝐹,𝑎   𝜑,𝑎,𝑏,𝑗   𝑌,𝑎,𝑏,𝑗
Allowed substitution hints:   𝜑(𝑥,𝑔,,𝑖)   𝐹(𝑥,𝑔,,𝑖,𝑗,𝑏)   𝑌(𝑥,𝑔,,𝑖)

Proof of Theorem fin23lem38
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 peano2 7128 . . . . . . . 8 (𝑑 ∈ ω → suc 𝑑 ∈ ω)
2 eqid 2651 . . . . . . . . . 10 ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)
3 fveq2 6229 . . . . . . . . . . . . . 14 (𝑏 = suc 𝑑 → (𝑌𝑏) = (𝑌‘suc 𝑑))
43rneqd 5385 . . . . . . . . . . . . 13 (𝑏 = suc 𝑑 → ran (𝑌𝑏) = ran (𝑌‘suc 𝑑))
54unieqd 4478 . . . . . . . . . . . 12 (𝑏 = suc 𝑑 ran (𝑌𝑏) = ran (𝑌‘suc 𝑑))
65eqeq2d 2661 . . . . . . . . . . 11 (𝑏 = suc 𝑑 → ( ran (𝑌‘suc 𝑑) = ran (𝑌𝑏) ↔ ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)))
76rspcev 3340 . . . . . . . . . 10 ((suc 𝑑 ∈ ω ∧ ran (𝑌‘suc 𝑑) = ran (𝑌‘suc 𝑑)) → ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
82, 7mpan2 707 . . . . . . . . 9 (suc 𝑑 ∈ ω → ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
9 fvex 6239 . . . . . . . . . . . 12 (𝑌‘suc 𝑑) ∈ V
109rnex 7142 . . . . . . . . . . 11 ran (𝑌‘suc 𝑑) ∈ V
1110uniex 6995 . . . . . . . . . 10 ran (𝑌‘suc 𝑑) ∈ V
12 eqid 2651 . . . . . . . . . . 11 (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = (𝑏 ∈ ω ↦ ran (𝑌𝑏))
1312elrnmpt 5404 . . . . . . . . . 10 ( ran (𝑌‘suc 𝑑) ∈ V → ( ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ↔ ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏)))
1411, 13ax-mp 5 . . . . . . . . 9 ( ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ↔ ∃𝑏 ∈ ω ran (𝑌‘suc 𝑑) = ran (𝑌𝑏))
158, 14sylibr 224 . . . . . . . 8 (suc 𝑑 ∈ ω → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
161, 15syl 17 . . . . . . 7 (𝑑 ∈ ω → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
1716adantl 481 . . . . . 6 ((𝜑𝑑 ∈ ω) → ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
18 intss1 4524 . . . . . 6 ( ran (𝑌‘suc 𝑑) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌‘suc 𝑑))
1917, 18syl 17 . . . . 5 ((𝜑𝑑 ∈ ω) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌‘suc 𝑑))
20 fin23lem33.f . . . . . 6 𝐹 = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
21 fin23lem.f . . . . . 6 (𝜑:ω–1-1→V)
22 fin23lem.g . . . . . 6 (𝜑 ran 𝐺)
23 fin23lem.h . . . . . 6 (𝜑 → ∀𝑗((𝑗:ω–1-1→V ∧ ran 𝑗𝐺) → ((𝑖𝑗):ω–1-1→V ∧ ran (𝑖𝑗) ⊊ ran 𝑗)))
24 fin23lem.i . . . . . 6 𝑌 = (rec(𝑖, ) ↾ ω)
2520, 21, 22, 23, 24fin23lem35 9207 . . . . 5 ((𝜑𝑑 ∈ ω) → ran (𝑌‘suc 𝑑) ⊊ ran (𝑌𝑑))
2619, 25sspsstrd 3748 . . . 4 ((𝜑𝑑 ∈ ω) → ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑))
27 dfpss2 3725 . . . . 5 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑) ↔ ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊆ ran (𝑌𝑑) ∧ ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑)))
2827simprbi 479 . . . 4 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ⊊ ran (𝑌𝑑) → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
2926, 28syl 17 . . 3 ((𝜑𝑑 ∈ ω) → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
3029nrexdv 3030 . 2 (𝜑 → ¬ ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
31 fveq2 6229 . . . . . . 7 (𝑏 = 𝑑 → (𝑌𝑏) = (𝑌𝑑))
3231rneqd 5385 . . . . . 6 (𝑏 = 𝑑 → ran (𝑌𝑏) = ran (𝑌𝑑))
3332unieqd 4478 . . . . 5 (𝑏 = 𝑑 ran (𝑌𝑏) = ran (𝑌𝑑))
3433cbvmptv 4783 . . . 4 (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = (𝑑 ∈ ω ↦ ran (𝑌𝑑))
3534elrnmpt 5404 . . 3 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) → ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ↔ ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑)))
3635ibi 256 . 2 ( ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) → ∃𝑑 ∈ ω ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) = ran (𝑌𝑑))
3730, 36nsyl 135 1 (𝜑 → ¬ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)) ∈ ran (𝑏 ∈ ω ↦ ran (𝑌𝑏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  wss 3607  wpss 3608  𝒫 cpw 4191   cuni 4468   cint 4507  cmpt 4762  ran crn 5144  cres 5145  suc csuc 5763  1-1wf1 5923  cfv 5926  (class class class)co 6690  ωcom 7107  reccrdg 7550  𝑚 cmap 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551
This theorem is referenced by:  fin23lem39  9210
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