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Theorem isf33lem 9586
Description: Lemma for isfin3-3 9588. (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 9585 . . . 4 (𝑓 ∈ FinIII → ¬ ω ≼* 𝑓)
2 fveq1 6498 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎‘suc 𝑥) = (𝑏‘suc 𝑥))
3 fveq1 6498 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎𝑥) = (𝑏𝑥))
42, 3sseq12d 3890 . . . . . . . . . 10 (𝑎 = 𝑏 → ((𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ (𝑏‘suc 𝑥) ⊆ (𝑏𝑥)))
54ralbidv 3147 . . . . . . . . 9 (𝑎 = 𝑏 → (∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) ↔ ∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥)))
6 rneq 5649 . . . . . . . . . . 11 (𝑎 = 𝑏 → ran 𝑎 = ran 𝑏)
76inteqd 4754 . . . . . . . . . 10 (𝑎 = 𝑏 ran 𝑎 = ran 𝑏)
87, 6eleq12d 2860 . . . . . . . . 9 (𝑎 = 𝑏 → ( ran 𝑎 ∈ ran 𝑎 ran 𝑏 ∈ ran 𝑏))
95, 8imbi12d 337 . . . . . . . 8 (𝑎 = 𝑏 → ((∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎) ↔ (∀𝑥 ∈ ω (𝑏‘suc 𝑥) ⊆ (𝑏𝑥) → ran 𝑏 ∈ ran 𝑏)))
109cbvralv 3383 . . . . . . 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 4425 . . . . . . . . 9 (𝑔 = 𝑦 → 𝒫 𝑔 = 𝒫 𝑦)
1211oveq1d 6991 . . . . . . . 8 (𝑔 = 𝑦 → (𝒫 𝑔𝑚 ω) = (𝒫 𝑦𝑚 ω))
1312raleqdv 3355 . . . . . . 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 275 . . . . . 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 2910 . . . . 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 9584 . . . 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 2844 . . . 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 9572 . . 3 (𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)} → 𝑓 ∈ FinIII)
2017, 19impbii 201 . 2 (𝑓 ∈ FinIII𝑓 ∈ {𝑔 ∣ ∀𝑎 ∈ (𝒫 𝑔𝑚 ω)(∀𝑥 ∈ ω (𝑎‘suc 𝑥) ⊆ (𝑎𝑥) → ran 𝑎 ∈ ran 𝑎)})
2120eqriv 2775 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 1507  wcel 2050  {cab 2758  wral 3088  wss 3829  𝒫 cpw 4422   cint 4749   class class class wbr 4929  ran crn 5408  suc csuc 6031  cfv 6188  (class class class)co 6976  ωcom 7396  𝑚 cmap 8206  * cwdom 8816  FinIIIcfin3 9501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-tr 5031  df-id 5312  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-se 5367  df-we 5368  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-pred 5986  df-ord 6032  df-on 6033  df-lim 6034  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-isom 6197  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-om 7397  df-1st 7501  df-2nd 7502  df-wrecs 7750  df-recs 7812  df-rdg 7850  df-seqom 7887  df-1o 7905  df-oadd 7909  df-er 8089  df-map 8208  df-en 8307  df-dom 8308  df-sdom 8309  df-fin 8310  df-wdom 8818  df-card 9162  df-fin4 9507  df-fin3 9508
This theorem is referenced by:  isfin3-2  9587  isfin3-3  9588  fin23  9609
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