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Theorem fin23lem39 10345
Description: Lemma for fin23 10384. 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 10344 . 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 10342 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ Ο‰) β†’ βˆͺ ran (π‘Œβ€˜suc 𝑒) ⊊ βˆͺ ran (π‘Œβ€˜π‘’))
87pssssd 4098 . . . . . 6 ((πœ‘ ∧ 𝑒 ∈ Ο‰) β†’ βˆͺ ran (π‘Œβ€˜suc 𝑒) βŠ† βˆͺ ran (π‘Œβ€˜π‘’))
9 peano2 7881 . . . . . . . . 9 (𝑒 ∈ Ο‰ β†’ suc 𝑒 ∈ Ο‰)
10 fveq2 6892 . . . . . . . . . . . 12 (𝑐 = suc 𝑒 β†’ (π‘Œβ€˜π‘) = (π‘Œβ€˜suc 𝑒))
1110rneqd 5938 . . . . . . . . . . 11 (𝑐 = suc 𝑒 β†’ ran (π‘Œβ€˜π‘) = ran (π‘Œβ€˜suc 𝑒))
1211unieqd 4923 . . . . . . . . . 10 (𝑐 = suc 𝑒 β†’ βˆͺ ran (π‘Œβ€˜π‘) = βˆͺ ran (π‘Œβ€˜suc 𝑒))
13 eqid 2733 . . . . . . . . . 10 (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))
14 fvex 6905 . . . . . . . . . . . 12 (π‘Œβ€˜suc 𝑒) ∈ V
1514rnex 7903 . . . . . . . . . . 11 ran (π‘Œβ€˜suc 𝑒) ∈ V
1615uniex 7731 . . . . . . . . . 10 βˆͺ ran (π‘Œβ€˜suc 𝑒) ∈ V
1712, 13, 16fvmpt 6999 . . . . . . . . 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 6892 . . . . . . . . . . 11 (𝑐 = 𝑒 β†’ (π‘Œβ€˜π‘) = (π‘Œβ€˜π‘’))
2019rneqd 5938 . . . . . . . . . 10 (𝑐 = 𝑒 β†’ ran (π‘Œβ€˜π‘) = ran (π‘Œβ€˜π‘’))
2120unieqd 4923 . . . . . . . . 9 (𝑐 = 𝑒 β†’ βˆͺ ran (π‘Œβ€˜π‘) = βˆͺ ran (π‘Œβ€˜π‘’))
22 fvex 6905 . . . . . . . . . . 11 (π‘Œβ€˜π‘’) ∈ V
2322rnex 7903 . . . . . . . . . 10 ran (π‘Œβ€˜π‘’) ∈ V
2423uniex 7731 . . . . . . . . 9 βˆͺ ran (π‘Œβ€˜π‘’) ∈ V
2521, 13, 24fvmpt 6999 . . . . . . . 8 (𝑒 ∈ Ο‰ β†’ ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’) = βˆͺ ran (π‘Œβ€˜π‘’))
2618, 25sseq12d 4016 . . . . . . 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 483 . . . . . 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 257 . . . . 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 3147 . . . 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 482 . . 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 6891 . . . . . . 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 6891 . . . . . . 7 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ (π‘‘β€˜π‘’) = ((𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘))β€˜π‘’))
3331, 32sseq12d 4016 . . . . . 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 3178 . . . . 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 5936 . . . . . . 7 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ ran 𝑑 = ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)))
3635inteqd 4956 . . . . . 6 (𝑑 = (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) β†’ ∩ ran 𝑑 = ∩ ran (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)))
3736, 35eleq12d 2828 . . . . 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 345 . . . 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 10324 . . . . . 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 267 . . . . 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 483 . . . 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 10341 . . . . . . . . 9 ((πœ‘ ∧ 𝑐 ∈ Ο‰) β†’ ((π‘Œβ€˜π‘):ω–1-1β†’V ∧ βˆͺ ran (π‘Œβ€˜π‘) βŠ† 𝐺))
4342simprd 497 . . . . . . . 8 ((πœ‘ ∧ 𝑐 ∈ Ο‰) β†’ βˆͺ ran (π‘Œβ€˜π‘) βŠ† 𝐺)
4443adantlr 714 . . . . . . 7 (((πœ‘ ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ Ο‰) β†’ βˆͺ ran (π‘Œβ€˜π‘) βŠ† 𝐺)
45 elpw2g 5345 . . . . . . . 8 (𝐺 ∈ 𝐹 β†’ (βˆͺ ran (π‘Œβ€˜π‘) ∈ 𝒫 𝐺 ↔ βˆͺ ran (π‘Œβ€˜π‘) βŠ† 𝐺))
4645ad2antlr 726 . . . . . . 7 (((πœ‘ ∧ 𝐺 ∈ 𝐹) ∧ 𝑐 ∈ Ο‰) β†’ (βˆͺ ran (π‘Œβ€˜π‘) ∈ 𝒫 𝐺 ↔ βˆͺ ran (π‘Œβ€˜π‘) βŠ† 𝐺))
4744, 46mpbird 257 . . . . . 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 5377 . . . . . 6 (𝐺 ∈ 𝐹 β†’ 𝒫 𝐺 ∈ V)
50 vex 3479 . . . . . . . 8 β„Ž ∈ V
51 f1f 6788 . . . . . . . 8 (β„Ž:ω–1-1β†’V β†’ β„Ž:Ο‰βŸΆV)
52 dmfex 7898 . . . . . . . 8 ((β„Ž ∈ V ∧ β„Ž:Ο‰βŸΆV) β†’ Ο‰ ∈ V)
5350, 51, 52sylancr 588 . . . . . . 7 (β„Ž:ω–1-1β†’V β†’ Ο‰ ∈ V)
542, 53syl 17 . . . . . 6 (πœ‘ β†’ Ο‰ ∈ V)
55 elmapg 8833 . . . . . 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 598 . . . . 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 257 . . . 4 ((πœ‘ ∧ 𝐺 ∈ 𝐹) β†’ (𝑐 ∈ Ο‰ ↦ βˆͺ ran (π‘Œβ€˜π‘)) ∈ (𝒫 𝐺 ↑m Ο‰))
5838, 41, 57rspcdva 3614 . . 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 815 1 (πœ‘ β†’ Β¬ 𝐺 ∈ 𝐹)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  Vcvv 3475   βŠ† wss 3949   ⊊ wpss 3950  π’« cpw 4603  βˆͺ cuni 4909  βˆ© cint 4951   ↦ cmpt 5232  ran crn 5678   β†Ύ cres 5679  suc csuc 6367  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€˜cfv 6544  (class class class)co 7409  Ο‰com 7855  reccrdg 8409   ↑m cmap 8820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-map 8822
This theorem is referenced by:  fin23lem41  10347
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