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Theorem isf32lem12 10385
Description: Lemma for isfin3-2 10388. (Contributed by Stefan O'Rear, 6-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
isf32lem40.f 𝐹 = {𝑔 ∣ βˆ€π‘Ž ∈ (𝒫 𝑔 ↑m Ο‰)(βˆ€π‘₯ ∈ Ο‰ (π‘Žβ€˜suc π‘₯) βŠ† (π‘Žβ€˜π‘₯) β†’ ∩ ran π‘Ž ∈ ran π‘Ž)}
Assertion
Ref Expression
isf32lem12 (𝐺 ∈ 𝑉 β†’ (Β¬ Ο‰ β‰Ό* 𝐺 β†’ 𝐺 ∈ 𝐹))
Distinct variable groups:   𝑔,𝐹   𝑔,π‘Ž,π‘₯,𝐺
Allowed substitution hints:   𝐹(π‘₯,π‘Ž)   𝑉(π‘₯,𝑔,π‘Ž)

Proof of Theorem isf32lem12
Dummy variables 𝑏 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elmapi 8864 . . . . 5 (𝑓 ∈ (𝒫 𝐺 ↑m Ο‰) β†’ 𝑓:Ο‰βŸΆπ’« 𝐺)
2 isf32lem11 10384 . . . . . . . . . 10 ((𝐺 ∈ 𝑉 ∧ (𝑓:Ο‰βŸΆπ’« 𝐺 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) ∧ Β¬ ∩ ran 𝑓 ∈ ran 𝑓)) β†’ Ο‰ β‰Ό* 𝐺)
32expcom 412 . . . . . . . . 9 ((𝑓:Ο‰βŸΆπ’« 𝐺 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) ∧ Β¬ ∩ ran 𝑓 ∈ ran 𝑓) β†’ (𝐺 ∈ 𝑉 β†’ Ο‰ β‰Ό* 𝐺))
433expa 1115 . . . . . . . 8 (((𝑓:Ο‰βŸΆπ’« 𝐺 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘)) ∧ Β¬ ∩ ran 𝑓 ∈ ran 𝑓) β†’ (𝐺 ∈ 𝑉 β†’ Ο‰ β‰Ό* 𝐺))
54impancom 450 . . . . . . 7 (((𝑓:Ο‰βŸΆπ’« 𝐺 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘)) ∧ 𝐺 ∈ 𝑉) β†’ (Β¬ ∩ ran 𝑓 ∈ ran 𝑓 β†’ Ο‰ β‰Ό* 𝐺))
65con1d 145 . . . . . 6 (((𝑓:Ο‰βŸΆπ’« 𝐺 ∧ βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘)) ∧ 𝐺 ∈ 𝑉) β†’ (Β¬ Ο‰ β‰Ό* 𝐺 β†’ ∩ ran 𝑓 ∈ ran 𝑓))
76exp31 418 . . . . 5 (𝑓:Ο‰βŸΆπ’« 𝐺 β†’ (βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ (𝐺 ∈ 𝑉 β†’ (Β¬ Ο‰ β‰Ό* 𝐺 β†’ ∩ ran 𝑓 ∈ ran 𝑓))))
81, 7syl 17 . . . 4 (𝑓 ∈ (𝒫 𝐺 ↑m Ο‰) β†’ (βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ (𝐺 ∈ 𝑉 β†’ (Β¬ Ο‰ β‰Ό* 𝐺 β†’ ∩ ran 𝑓 ∈ ran 𝑓))))
98com4t 93 . . 3 (𝐺 ∈ 𝑉 β†’ (Β¬ Ο‰ β‰Ό* 𝐺 β†’ (𝑓 ∈ (𝒫 𝐺 ↑m Ο‰) β†’ (βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ ∩ ran 𝑓 ∈ ran 𝑓))))
109ralrimdv 3142 . 2 (𝐺 ∈ 𝑉 β†’ (Β¬ Ο‰ β‰Ό* 𝐺 β†’ βˆ€π‘“ ∈ (𝒫 𝐺 ↑m Ο‰)(βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ ∩ ran 𝑓 ∈ ran 𝑓)))
11 isf32lem40.f . . 3 𝐹 = {𝑔 ∣ βˆ€π‘Ž ∈ (𝒫 𝑔 ↑m Ο‰)(βˆ€π‘₯ ∈ Ο‰ (π‘Žβ€˜suc π‘₯) βŠ† (π‘Žβ€˜π‘₯) β†’ ∩ ran π‘Ž ∈ ran π‘Ž)}
1211isfin3ds 10350 . 2 (𝐺 ∈ 𝑉 β†’ (𝐺 ∈ 𝐹 ↔ βˆ€π‘“ ∈ (𝒫 𝐺 ↑m Ο‰)(βˆ€π‘ ∈ Ο‰ (π‘“β€˜suc 𝑏) βŠ† (π‘“β€˜π‘) β†’ ∩ ran 𝑓 ∈ ran 𝑓)))
1310, 12sylibrd 258 1 (𝐺 ∈ 𝑉 β†’ (Β¬ Ο‰ β‰Ό* 𝐺 β†’ 𝐺 ∈ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2702  βˆ€wral 3051   βŠ† wss 3940  π’« cpw 4598  βˆ© cint 4944   class class class wbr 5143  ran crn 5673  suc csuc 6366  βŸΆwf 6538  β€˜cfv 6542  (class class class)co 7415  Ο‰com 7867   ↑m cmap 8841   β‰Ό* cwdom 9585
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-1o 8483  df-er 8721  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-fin 8964  df-wdom 9586  df-card 9960
This theorem is referenced by:  isf33lem  10387  isfin3-2  10388
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