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Theorem fin23lem39 10347
Description: Lemma for fin23 10386. Thus, we have that 𝑔 could not have been in 𝐹 after all. (Contributed by Stefan O'Rear, 4-Nov-2014.)
Hypotheses
Ref Expression
fin23lem33.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)}
fin23lem.f (πœ‘ β†’ β„Ž:ω–1-1β†’V)
fin23lem.g (πœ‘ β†’ βˆͺ ran β„Ž βŠ† 𝐺)
fin23lem.h (πœ‘ β†’ βˆ€π‘—((𝑗:ω–1-1β†’V ∧ βˆͺ ran 𝑗 βŠ† 𝐺) β†’ ((π‘–β€˜π‘—):ω–1-1β†’V ∧ βˆͺ ran (π‘–β€˜π‘—) ⊊ βˆͺ ran 𝑗)))
fin23lem.i π‘Œ = (rec(𝑖, β„Ž) β†Ύ Ο‰)
Assertion
Ref Expression
fin23lem39 (πœ‘ β†’ Β¬ 𝐺 ∈ 𝐹)
Distinct variable groups:   𝑔,π‘Ž,𝑖,𝑗,π‘₯,β„Ž,𝐺   𝐹,π‘Ž   πœ‘,π‘Ž,𝑗   π‘Œ,π‘Ž,𝑗
Allowed substitution hints:   πœ‘(π‘₯,𝑔,β„Ž,𝑖)   𝐹(π‘₯,𝑔,β„Ž,𝑖,𝑗)   π‘Œ(π‘₯,𝑔,β„Ž,𝑖)
Proof of Theorem fin23lem39
Dummy variables 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fin23lem33.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)}
2 fin23lem.f . . 3 (πœ‘ β†’ β„Ž:ω–1-1β†’V)
3 fin23lem.g . . 3 (πœ‘ β†’ βˆͺ ran β„Ž βŠ† 𝐺)
4 fin23lem.h . . 3 (πœ‘ β†’ βˆ€π‘—((𝑗:ω–1-1β†’V ∧ βˆͺ ran 𝑗 βŠ† 𝐺) β†’ ((π‘–β€˜π‘—):ω–1-1β†’V ∧ βˆͺ ran (π‘–β€˜π‘—) ⊊ βˆͺ ran 𝑗)))
5 fin23lem.i . . 3 π‘Œ = (rec(𝑖, β„Ž) β†Ύ Ο‰)
61, 2, 3, 4, 5fin23lem38 10346 . 2 (πœ‘ β†’ Β¬ ∩ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) ∈ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)))
71, 2, 3, 4, 5fin23lem35 10344 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ Ο‰) β†’ βˆͺ ran (π‘Œβ€˜suc 𝑒) ⊊ βˆͺ ran (π‘Œβ€˜π‘’))
87pssssd 4096 . . . . . 6 ((πœ‘ ∧ 𝑒 ∈ Ο‰) β†’ βˆͺ ran (π‘Œβ€˜suc 𝑒) βŠ† βˆͺ ran (π‘Œβ€˜π‘’))
9 peano2 7883 . . . . . . . . 9 (𝑒 ∈ Ο‰ β†’ suc 𝑒 ∈ Ο‰)
10 fveq2 6890 . . . . . . . . . . . 12 (𝑐 = suc 𝑒 β†’ (π‘Œβ€˜π‘) = (π‘Œβ€˜suc 𝑒))
1110rneqd 5936 . . . . . . . . . . 11 (𝑐 = suc 𝑒 β†’ ran (π‘Œβ€˜π‘) = ran (π‘Œβ€˜suc 𝑒))
1211unieqd 4921 . . . . . . . . . 10 (𝑐 = suc 𝑒 β†’ βˆͺ ran (π‘Œβ€˜π‘) = βˆͺ ran (π‘Œβ€˜suc 𝑒))
13 eqid 2730 . . . . . . . . . 10 (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))
14 fvex 6903 . . . . . . . . . . . 12 (π‘Œβ€˜suc 𝑒) ∈ V
1514rnex 7905 . . . . . . . . . . 11 ran (π‘Œβ€˜suc 𝑒) ∈ V
1615uniex 7733 . . . . . . . . . 10 βˆͺ ran (π‘Œβ€˜suc 𝑒) ∈ V
1712, 13, 16fvmpt 6997 . . . . . . . . 9 (suc 𝑒 ∈ Ο‰ β†’ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) = βˆͺ ran (π‘Œβ€˜suc 𝑒))
189, 17syl 17 . . . . . . . 8 (𝑒 ∈ Ο‰ β†’ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) = βˆͺ ran (π‘Œβ€˜suc 𝑒))
19 fveq2 6890 . . . . . . . . . . 11 (𝑐 = 𝑒 β†’ (π‘Œβ€˜π‘) = (π‘Œβ€˜π‘’))
2019rneqd 5936 . . . . . . . . . 10 (𝑐 = 𝑒 β†’ ran (π‘Œβ€˜π‘) = ran (π‘Œβ€˜π‘’))
2120unieqd 4921 . . . . . . . . 9 (𝑐 = 𝑒 β†’ βˆͺ ran (π‘Œβ€˜π‘) = βˆͺ ran (π‘Œβ€˜π‘’))
22 fvex 6903 . . . . . . . . . . 11 (π‘Œβ€˜π‘’) ∈ V
2322rnex 7905 . . . . . . . . . 10 ran (π‘Œβ€˜π‘’) ∈ V
2423uniex 7733 . . . . . . . . 9 βˆͺ ran (π‘Œβ€˜π‘’) ∈ V
2521, 13, 24fvmpt 6997 . . . . . . . 8 (𝑒 ∈ Ο‰ β†’ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’) = βˆͺ ran (π‘Œβ€˜π‘’))
2618, 25sseq12d 4014 . . . . . . 7 (𝑒 ∈ Ο‰ β†’ (((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) βŠ† ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’) ↔ βˆͺ ran (π‘Œβ€˜suc 𝑒) βŠ† βˆͺ ran (π‘Œβ€˜π‘’)))
2726adantl 480 . . . . . 6 ((πœ‘ ∧ 𝑒 ∈ Ο‰) β†’ (((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) βŠ† ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’) ↔ βˆͺ ran (π‘Œβ€˜suc 𝑒) βŠ† βˆͺ ran (π‘Œβ€˜π‘’)))
288, 27mpbird 256 . . . . 5 ((πœ‘ ∧ 𝑒 ∈ Ο‰) β†’ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) βŠ† ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’))
2928ralrimiva 3144 . . . 4 (πœ‘ β†’ βˆ€π‘’ ∈ Ο‰ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) βŠ† ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’))
3029adantr 479 . . 3 ((πœ‘ ∧ 𝐺 ∈ 𝐹) β†’ βˆ€π‘’ ∈ Ο‰ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) βŠ† ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’))
31 fveq1 6889 . . . . . . 7 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ (π‘‘β€˜suc 𝑒) = ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒))
32 fveq1 6889 . . . . . . 7 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ (π‘‘β€˜π‘’) = ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’))
3331, 32sseq12d 4014 . . . . . 6 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ ((π‘‘β€˜suc 𝑒) βŠ† (π‘‘β€˜π‘’) ↔ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) βŠ† ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’)))
3433ralbidv 3175 . . . . 5 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ (βˆ€π‘’ ∈ Ο‰ (π‘‘β€˜suc 𝑒) βŠ† (π‘‘β€˜π‘’) ↔ βˆ€π‘’ ∈ Ο‰ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) βŠ† ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’)))
35 rneq 5934 . . . . . . 7 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ ran 𝑑 = ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)))
3635inteqd 4954 . . . . . 6 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ ∩ ran 𝑑 = ∩ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)))
3736, 35eleq12d 2825 . . . . 5 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ (∩ ran 𝑑 ∈ ran 𝑑 ↔ ∩ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) ∈ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))))
3834, 37imbi12d 343 . . . 4 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ ((βˆ€π‘’ ∈ Ο‰ (π‘‘β€˜suc 𝑒) βŠ† (π‘‘β€˜π‘’) β†’ ∩ ran 𝑑 ∈ ran 𝑑) ↔ (βˆ€π‘’ ∈ Ο‰ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) βŠ† ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’) β†’ ∩ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) ∈ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)))))
391isfin3ds 10326 . . . . . 6 (𝐺 ∈ 𝐹 β†’ (𝐺 ∈ 𝐹 ↔ βˆ€π‘‘ ∈ (𝒫 𝐺 ↑m Ο‰)(βˆ€π‘’ ∈ Ο‰ (π‘‘β€˜suc 𝑒) βŠ† (π‘‘β€˜π‘’) β†’ ∩ ran 𝑑 ∈ ran 𝑑)))
4039ibi 266 . . . . 5 (𝐺 ∈ 𝐹 β†’ βˆ€π‘‘ ∈ (𝒫 𝐺 ↑m Ο‰)(βˆ€π‘’ ∈ Ο‰ (π‘‘β€˜suc 𝑒) βŠ† (π‘‘β€˜π‘’) β†’ ∩ ran 𝑑 ∈ ran 𝑑))
4140adantl 480 . . . 4 ((πœ‘ ∧ 𝐺 ∈ 𝐹) β†’ βˆ€π‘‘ ∈ (𝒫 𝐺 ↑m Ο‰)(βˆ€π‘’ ∈ Ο‰ (π‘‘β€˜suc 𝑒) βŠ† (π‘‘β€˜π‘’) β†’ ∩ ran 𝑑 ∈ ran 𝑑))
421, 2, 3, 4, 5fin23lem34 10343 . . . . . . . . 9 ((πœ‘ ∧ 𝑐 ∈ Ο‰) β†’ ((π‘Œβ€˜π‘):ω–1-1β†’V ∧ βˆͺ ran (π‘Œβ€˜π‘) βŠ† 𝐺))
4342simprd 494 . . . . . . . 8 ((πœ‘ ∧ 𝑐 ∈ Ο‰) β†’ βˆͺ ran (π‘Œβ€˜π‘) βŠ† 𝐺)
4443adantlr 711 . . . . . . 7 (((πœ‘ ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ Ο‰) β†’ βˆͺ ran (π‘Œβ€˜π‘) βŠ† 𝐺)
45 elpw2g 5343 . . . . . . . 8 (𝐺 ∈ 𝐹 β†’ (βˆͺ ran (π‘Œβ€˜π‘) ∈ 𝒫 𝐺 ↔ βˆͺ ran (π‘Œβ€˜π‘) βŠ† 𝐺))
4645ad2antlr 723 . . . . . . 7 (((πœ‘ ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ Ο‰) β†’ (βˆͺ ran (π‘Œβ€˜π‘) ∈ 𝒫 𝐺 ↔ βˆͺ ran (π‘Œβ€˜π‘) βŠ† 𝐺))
4744, 46mpbird 256 . . . . . 6 (((πœ‘ ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ Ο‰) β†’ βˆͺ ran (π‘Œβ€˜π‘) ∈ 𝒫 𝐺)
4847fmpttd 7115 . . . . 5 ((πœ‘ ∧ 𝐺 ∈ 𝐹) β†’ (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)):Ο‰βŸΆπ’« 𝐺)
49 pwexg 5375 . . . . . 6 (𝐺 ∈ 𝐹 β†’ 𝒫 𝐺 ∈ V)
50 vex 3476 . . . . . . . 8 β„Ž ∈ V
51 f1f 6786 . . . . . . . 8 (β„Ž:ω–1-1β†’V β†’ β„Ž:Ο‰βŸΆV)
52 dmfex 7900 . . . . . . . 8 ((β„Ž ∈ V ∧ β„Ž:Ο‰βŸΆV) β†’ Ο‰ ∈ V)
5350, 51, 52sylancr 585 . . . . . . 7 (β„Ž:ω–1-1β†’V β†’ Ο‰ ∈ V)
542, 53syl 17 . . . . . 6 (πœ‘ β†’ Ο‰ ∈ V)
55 elmapg 8835 . . . . . 6 ((𝒫 𝐺 ∈ V ∧ Ο‰ ∈ V) β†’ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) ∈ (𝒫 𝐺 ↑m Ο‰) ↔ (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)):Ο‰βŸΆπ’« 𝐺))
5649, 54, 55syl2anr 595 . . . . 5 ((πœ‘ ∧ 𝐺 ∈ 𝐹) β†’ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) ∈ (𝒫 𝐺 ↑m Ο‰) ↔ (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)):Ο‰βŸΆπ’« 𝐺))
5748, 56mpbird 256 . . . 4 ((πœ‘ ∧ 𝐺 ∈ 𝐹) β†’ (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) ∈ (𝒫 𝐺 ↑m Ο‰))
5838, 41, 57rspcdva 3612 . . 3 ((πœ‘ ∧ 𝐺 ∈ 𝐹) β†’ (βˆ€π‘’ ∈ Ο‰ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜suc 𝑒) βŠ† ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’) β†’ ∩ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) ∈ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))))
5930, 58mpd 15 . 2 ((πœ‘ ∧ 𝐺 ∈ 𝐹) β†’ ∩ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) ∈ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)))
606, 59mtand 812 1 (πœ‘ β†’ Β¬ 𝐺 ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394  βˆ€wal 1537   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆ€wral 3059  Vcvv 3472   βŠ† wss 3947   ⊊ wpss 3948  π’« cpw 4601  βˆͺ cuni 4907  βˆ© cint 4949   ↦ cmpt 5230  ran crn 5676   β†Ύ cres 5677  suc csuc 6365  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7411  Ο‰com 7857  reccrdg 8411   ↑m cmap 8822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  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-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-map 8824
This theorem is referenced by:  fin23lem41  10349
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