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Theorem isf33lem 10399
Description: Lemma for isfin3-3 10401. (Contributed by Stefan O'Rear, 17-May-2015.)
Assertion
Ref Expression
isf33lem FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
Distinct variable group:   𝑔,𝑎,𝑥

Proof of Theorem isf33lem
Dummy variables 𝑏 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfin32i 10398 . . . 4 (𝑓 ∈ FinIII → ¬ ω ≼* 𝑓)
2 fveq1 6901 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎‘suc 𝑥) = (𝑏‘suc 𝑥))
3 fveq1 6901 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎𝑥) = (𝑏𝑥))
42, 3sseq12d 4015 . . . . . . . . . 10 (𝑎 = 𝑏 → ((𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ (𝑏‘suc 𝑥) ⊆ (𝑏𝑥)))
54ralbidv 3175 . . . . . . . . 9 (𝑎 = 𝑏 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ ∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥)))
6 rneq 5942 . . . . . . . . . . 11 (𝑎 = 𝑏 → ran 𝑎 = ran 𝑏)
76inteqd 4958 . . . . . . . . . 10 (𝑎 = 𝑏 ran 𝑎 = ran 𝑏)
87, 6eleq12d 2823 . . . . . . . . 9 (𝑎 = 𝑏 → ( ran 𝑎 ∈ ran 𝑎 ran 𝑏 ∈ ran 𝑏))
95, 8imbi12d 343 . . . . . . . 8 (𝑎 = 𝑏 → ((∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏)))
109cbvralvw 3232 . . . . . . 7 (∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎) ↔ ∀𝑏 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏))
11 pweq 4620 . . . . . . . . 9 (𝑔 = 𝑦 → 𝒫 𝑔 = 𝒫 𝑦)
1211oveq1d 7441 . . . . . . . 8 (𝑔 = 𝑦 → (𝒫 𝑔m ω) = (𝒫 𝑦m ω))
1312raleqdv 3323 . . . . . . 7 (𝑔 = 𝑦 → (∀𝑏 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏) ↔ ∀𝑏 ∈ (𝒫 𝑦m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏)))
1410, 13bitrid 282 . . . . . 6 (𝑔 = 𝑦 → (∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎) ↔ ∀𝑏 ∈ (𝒫 𝑦m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏)))
1514cbvabv 2801 . . . . 5 {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)} = {𝑦 ∣ ∀𝑏 ∈ (𝒫 𝑦m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏)}
1615isf32lem12 10397 . . . 4 (𝑓 ∈ FinIII → (¬ ω ≼* 𝑓𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}))
171, 16mpd 15 . . 3 (𝑓 ∈ FinIII𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)})
1810abbii 2798 . . . 4 {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)} = {𝑔 ∣ ∀𝑏 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏)}
1918fin23lem41 10385 . . 3 (𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)} → 𝑓 ∈ FinIII)
2017, 19impbii 208 . 2 (𝑓 ∈ FinIII𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)})
2120eqriv 2725 1 FinIII = {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  {cab 2705  wral 3058  wss 3949  𝒫 cpw 4606   cint 4953   class class class wbr 5152  ran crn 5683  suc csuc 6376  cfv 6553  (class class class)co 7426  ωcom 7878  m cmap 8853  * cwdom 9597  FinIIIcfin3 10314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-isom 6562  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-seqom 8477  df-1o 8495  df-er 8733  df-map 8855  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-wdom 9598  df-card 9972  df-fin4 10320  df-fin3 10321
This theorem is referenced by:  isfin3-2  10400  isfin3-3  10401  fin23  10422
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