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Theorem isf32lem12 10361
Description: Lemma for isfin3-2 10364. (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 8845 . . . . 5 (𝑓 ∈ (𝒫 𝐺 ↑m Ο‰) β†’ 𝑓:Ο‰βŸΆπ’« 𝐺)
2 isf32lem11 10360 . . . . . . . . . 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 413 . . . . . . . . 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 451 . . . . . . 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 419 . . . . 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 3146 . 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 10326 . 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 259 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆ€wral 3055   βŠ† wss 3943  π’« cpw 4597  βˆ© cint 4943   class class class wbr 5141  ran crn 5670  suc csuc 6360  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405  Ο‰com 7852   ↑m cmap 8822   β‰Ό* cwdom 9561
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-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
This theorem is referenced by:  isf33lem  10363  isfin3-2  10364
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