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Theorem isf33lem 10363
Description: Lemma for isfin3-3 10365. (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 10362 . . . 4 (𝑓 ∈ FinIII → ¬ ω ≼* 𝑓)
2 fveq1 6884 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎‘suc 𝑥) = (𝑏‘suc 𝑥))
3 fveq1 6884 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎𝑥) = (𝑏𝑥))
42, 3sseq12d 4010 . . . . . . . . . 10 (𝑎 = 𝑏 → ((𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ (𝑏‘suc 𝑥) ⊆ (𝑏𝑥)))
54ralbidv 3171 . . . . . . . . 9 (𝑎 = 𝑏 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ ∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥)))
6 rneq 5929 . . . . . . . . . . 11 (𝑎 = 𝑏 → ran 𝑎 = ran 𝑏)
76inteqd 4948 . . . . . . . . . 10 (𝑎 = 𝑏 ran 𝑎 = ran 𝑏)
87, 6eleq12d 2821 . . . . . . . . 9 (𝑎 = 𝑏 → ( ran 𝑎 ∈ ran 𝑎 ran 𝑏 ∈ ran 𝑏))
95, 8imbi12d 344 . . . . . . . 8 (𝑎 = 𝑏 → ((∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏)))
109cbvralvw 3228 . . . . . . 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 4611 . . . . . . . . 9 (𝑔 = 𝑦 → 𝒫 𝑔 = 𝒫 𝑦)
1211oveq1d 7420 . . . . . . . 8 (𝑔 = 𝑦 → (𝒫 𝑔m ω) = (𝒫 𝑦m ω))
1312raleqdv 3319 . . . . . . 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 283 . . . . . 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 2799 . . . . 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 10361 . . . 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 2796 . . . 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 10349 . . 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 2723 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 2703  wral 3055  wss 3943  𝒫 cpw 4597   cint 4943   class class class wbr 5141  ran crn 5670  suc csuc 6360  cfv 6537  (class class class)co 7405  ωcom 7852  m cmap 8822  * cwdom 9561  FinIIIcfin3 10278
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-seqom 8449  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-wdom 9562  df-card 9936  df-fin4 10284  df-fin3 10285
This theorem is referenced by:  isfin3-2  10364  isfin3-3  10365  fin23  10386
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