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Theorem isf33lem 9132
Description: Lemma for isfin3-3 9134. (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 9131 . . . 4 (𝑓 ∈ FinIII → ¬ ω ≼* 𝑓)
2 fveq1 6147 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎‘suc 𝑥) = (𝑏‘suc 𝑥))
3 fveq1 6147 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎𝑥) = (𝑏𝑥))
42, 3sseq12d 3613 . . . . . . . . . 10 (𝑎 = 𝑏 → ((𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ (𝑏‘suc 𝑥) ⊆ (𝑏𝑥)))
54ralbidv 2980 . . . . . . . . 9 (𝑎 = 𝑏 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ ∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥)))
6 rneq 5311 . . . . . . . . . . 11 (𝑎 = 𝑏 → ran 𝑎 = ran 𝑏)
76inteqd 4445 . . . . . . . . . 10 (𝑎 = 𝑏 ran 𝑎 = ran 𝑏)
87, 6eleq12d 2692 . . . . . . . . 9 (𝑎 = 𝑏 → ( ran 𝑎 ∈ ran 𝑎 ran 𝑏 ∈ ran 𝑏))
95, 8imbi12d 334 . . . . . . . 8 (𝑎 = 𝑏 → ((∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏)))
109cbvralv 3159 . . . . . . 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 4133 . . . . . . . . 9 (𝑔 = 𝑦 → 𝒫 𝑔 = 𝒫 𝑦)
1211oveq1d 6619 . . . . . . . 8 (𝑔 = 𝑦 → (𝒫 𝑔𝑚 ω) = (𝒫 𝑦𝑚 ω))
1312raleqdv 3133 . . . . . . 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 272 . . . . . 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 2744 . . . . 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 9130 . . . 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 2736 . . . 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 9118 . . 3 (𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)} → 𝑓 ∈ FinIII)
2017, 19impbii 199 . 2 (𝑓 ∈ FinIII𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)})
2120eqriv 2618 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 1480  wcel 1987  {cab 2607  wral 2907  wss 3555  𝒫 cpw 4130   cint 4440   class class class wbr 4613  ran crn 5075  suc csuc 5684  cfv 5847  (class class class)co 6604  ωcom 7012  𝑚 cmap 7802  * cwdom 8406  FinIIIcfin3 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seqom 7488  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-wdom 8408  df-card 8709  df-fin4 9053  df-fin3 9054
This theorem is referenced by:  isfin3-2  9133  isfin3-3  9134  fin23  9155
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