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Theorem isf33lem 9786
 Description: Lemma for isfin3-3 9788. (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 9785 . . . 4 (𝑓 ∈ FinIII → ¬ ω ≼* 𝑓)
2 fveq1 6660 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎‘suc 𝑥) = (𝑏‘suc 𝑥))
3 fveq1 6660 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎𝑥) = (𝑏𝑥))
42, 3sseq12d 3986 . . . . . . . . . 10 (𝑎 = 𝑏 → ((𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ (𝑏‘suc 𝑥) ⊆ (𝑏𝑥)))
54ralbidv 3192 . . . . . . . . 9 (𝑎 = 𝑏 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ ∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥)))
6 rneq 5793 . . . . . . . . . . 11 (𝑎 = 𝑏 → ran 𝑎 = ran 𝑏)
76inteqd 4867 . . . . . . . . . 10 (𝑎 = 𝑏 ran 𝑎 = ran 𝑏)
87, 6eleq12d 2910 . . . . . . . . 9 (𝑎 = 𝑏 → ( ran 𝑎 ∈ ran 𝑎 ran 𝑏 ∈ ran 𝑏))
95, 8imbi12d 348 . . . . . . . 8 (𝑎 = 𝑏 → ((∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏)))
109cbvralvw 3434 . . . . . . 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 4538 . . . . . . . . 9 (𝑔 = 𝑦 → 𝒫 𝑔 = 𝒫 𝑦)
1211oveq1d 7164 . . . . . . . 8 (𝑔 = 𝑦 → (𝒫 𝑔m ω) = (𝒫 𝑦m ω))
1312raleqdv 3402 . . . . . . 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, 13syl5bb 286 . . . . . 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 2892 . . . . 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 9784 . . . 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 2889 . . . 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 9772 . . 3 (𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)} → 𝑓 ∈ FinIII)
2017, 19impbii 212 . 2 (𝑓 ∈ FinIII𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔m ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)})
2120eqriv 2821 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 1538   ∈ wcel 2115  {cab 2802  ∀wral 3133   ⊆ wss 3919  𝒫 cpw 4522  ∩ cint 4862   class class class wbr 5052  ran crn 5543  suc csuc 6180  ‘cfv 6343  (class class class)co 7149  ωcom 7574   ↑m cmap 8402   ≼* cwdom 9025  FinIIIcfin3 9701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-seqom 8080  df-1o 8098  df-oadd 8102  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-wdom 9026  df-card 9365  df-fin4 9707  df-fin3 9708 This theorem is referenced by:  isfin3-2  9787  isfin3-3  9788  fin23  9809
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