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Theorem fin23lem21 10336
Description: Lemma for fin23 10386. 𝑋 is not empty. We only need here that 𝑑 has at least one set in its range besides βˆ…; the much stronger hypothesis here will serve as our induction hypothesis though. (Contributed by Stefan O'Rear, 1-Nov-2014.) (Revised by Mario Carneiro, 6-May-2015.)
Hypotheses
Ref Expression
fin23lem.a π‘ˆ = seqΟ‰((𝑖 ∈ Ο‰, 𝑒 ∈ V ↦ if(((π‘‘β€˜π‘–) ∩ 𝑒) = βˆ…, 𝑒, ((π‘‘β€˜π‘–) ∩ 𝑒))), βˆͺ ran 𝑑)
fin23lem17.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
fin23lem21 ((βˆͺ ran 𝑑 ∈ 𝐹 ∧ 𝑑:ω–1-1→𝑉) β†’ ∩ ran π‘ˆ β‰  βˆ…)
Distinct variable groups:   𝑔,𝑖,𝑑,𝑒,π‘₯,π‘Ž   𝐹,π‘Ž,𝑑   𝑉,π‘Ž   π‘₯,π‘Ž   π‘ˆ,π‘Ž,𝑖,𝑒   𝑔,π‘Ž
Allowed substitution hints:   π‘ˆ(π‘₯,𝑑,𝑔)   𝐹(π‘₯,𝑒,𝑔,𝑖)   𝑉(π‘₯,𝑒,𝑑,𝑔,𝑖)

Proof of Theorem fin23lem21
StepHypRef Expression
1 fin23lem.a . . 3 π‘ˆ = seqΟ‰((𝑖 ∈ Ο‰, 𝑒 ∈ V ↦ if(((π‘‘β€˜π‘–) ∩ 𝑒) = βˆ…, 𝑒, ((π‘‘β€˜π‘–) ∩ 𝑒))), βˆͺ ran 𝑑)
2 fin23lem17.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)}
31, 2fin23lem17 10335 . 2 ((βˆͺ ran 𝑑 ∈ 𝐹 ∧ 𝑑:ω–1-1→𝑉) β†’ ∩ ran π‘ˆ ∈ ran π‘ˆ)
41fnseqom 8456 . . . . 5 π‘ˆ Fn Ο‰
5 fvelrnb 6946 . . . . 5 (π‘ˆ Fn Ο‰ β†’ (∩ ran π‘ˆ ∈ ran π‘ˆ ↔ βˆƒπ‘Ž ∈ Ο‰ (π‘ˆβ€˜π‘Ž) = ∩ ran π‘ˆ))
64, 5ax-mp 5 . . . 4 (∩ ran π‘ˆ ∈ ran π‘ˆ ↔ βˆƒπ‘Ž ∈ Ο‰ (π‘ˆβ€˜π‘Ž) = ∩ ran π‘ˆ)
7 id 22 . . . . . . 7 (π‘Ž ∈ Ο‰ β†’ π‘Ž ∈ Ο‰)
8 vex 3472 . . . . . . . . . 10 𝑑 ∈ V
9 f1f1orn 6838 . . . . . . . . . 10 (𝑑:ω–1-1→𝑉 β†’ 𝑑:ω–1-1-ontoβ†’ran 𝑑)
10 f1oen3g 8964 . . . . . . . . . 10 ((𝑑 ∈ V ∧ 𝑑:ω–1-1-ontoβ†’ran 𝑑) β†’ Ο‰ β‰ˆ ran 𝑑)
118, 9, 10sylancr 586 . . . . . . . . 9 (𝑑:ω–1-1→𝑉 β†’ Ο‰ β‰ˆ ran 𝑑)
12 ominf 9260 . . . . . . . . 9 Β¬ Ο‰ ∈ Fin
13 ssdif0 4358 . . . . . . . . . . 11 (ran 𝑑 βŠ† {βˆ…} ↔ (ran 𝑑 βˆ– {βˆ…}) = βˆ…)
14 snfi 9046 . . . . . . . . . . . . 13 {βˆ…} ∈ Fin
15 ssfi 9175 . . . . . . . . . . . . 13 (({βˆ…} ∈ Fin ∧ ran 𝑑 βŠ† {βˆ…}) β†’ ran 𝑑 ∈ Fin)
1614, 15mpan 687 . . . . . . . . . . . 12 (ran 𝑑 βŠ† {βˆ…} β†’ ran 𝑑 ∈ Fin)
17 enfi 9192 . . . . . . . . . . . 12 (Ο‰ β‰ˆ ran 𝑑 β†’ (Ο‰ ∈ Fin ↔ ran 𝑑 ∈ Fin))
1816, 17imbitrrid 245 . . . . . . . . . . 11 (Ο‰ β‰ˆ ran 𝑑 β†’ (ran 𝑑 βŠ† {βˆ…} β†’ Ο‰ ∈ Fin))
1913, 18biimtrrid 242 . . . . . . . . . 10 (Ο‰ β‰ˆ ran 𝑑 β†’ ((ran 𝑑 βˆ– {βˆ…}) = βˆ… β†’ Ο‰ ∈ Fin))
2019necon3bd 2948 . . . . . . . . 9 (Ο‰ β‰ˆ ran 𝑑 β†’ (Β¬ Ο‰ ∈ Fin β†’ (ran 𝑑 βˆ– {βˆ…}) β‰  βˆ…))
2111, 12, 20mpisyl 21 . . . . . . . 8 (𝑑:ω–1-1→𝑉 β†’ (ran 𝑑 βˆ– {βˆ…}) β‰  βˆ…)
22 n0 4341 . . . . . . . . 9 ((ran 𝑑 βˆ– {βˆ…}) β‰  βˆ… ↔ βˆƒπ‘Ž π‘Ž ∈ (ran 𝑑 βˆ– {βˆ…}))
23 eldifsn 4785 . . . . . . . . . . 11 (π‘Ž ∈ (ran 𝑑 βˆ– {βˆ…}) ↔ (π‘Ž ∈ ran 𝑑 ∧ π‘Ž β‰  βˆ…))
24 elssuni 4934 . . . . . . . . . . . 12 (π‘Ž ∈ ran 𝑑 β†’ π‘Ž βŠ† βˆͺ ran 𝑑)
25 ssn0 4395 . . . . . . . . . . . 12 ((π‘Ž βŠ† βˆͺ ran 𝑑 ∧ π‘Ž β‰  βˆ…) β†’ βˆͺ ran 𝑑 β‰  βˆ…)
2624, 25sylan 579 . . . . . . . . . . 11 ((π‘Ž ∈ ran 𝑑 ∧ π‘Ž β‰  βˆ…) β†’ βˆͺ ran 𝑑 β‰  βˆ…)
2723, 26sylbi 216 . . . . . . . . . 10 (π‘Ž ∈ (ran 𝑑 βˆ– {βˆ…}) β†’ βˆͺ ran 𝑑 β‰  βˆ…)
2827exlimiv 1925 . . . . . . . . 9 (βˆƒπ‘Ž π‘Ž ∈ (ran 𝑑 βˆ– {βˆ…}) β†’ βˆͺ ran 𝑑 β‰  βˆ…)
2922, 28sylbi 216 . . . . . . . 8 ((ran 𝑑 βˆ– {βˆ…}) β‰  βˆ… β†’ βˆͺ ran 𝑑 β‰  βˆ…)
3021, 29syl 17 . . . . . . 7 (𝑑:ω–1-1→𝑉 β†’ βˆͺ ran 𝑑 β‰  βˆ…)
311fin23lem14 10330 . . . . . . 7 ((π‘Ž ∈ Ο‰ ∧ βˆͺ ran 𝑑 β‰  βˆ…) β†’ (π‘ˆβ€˜π‘Ž) β‰  βˆ…)
327, 30, 31syl2anr 596 . . . . . 6 ((𝑑:ω–1-1→𝑉 ∧ π‘Ž ∈ Ο‰) β†’ (π‘ˆβ€˜π‘Ž) β‰  βˆ…)
33 neeq1 2997 . . . . . 6 ((π‘ˆβ€˜π‘Ž) = ∩ ran π‘ˆ β†’ ((π‘ˆβ€˜π‘Ž) β‰  βˆ… ↔ ∩ ran π‘ˆ β‰  βˆ…))
3432, 33syl5ibcom 244 . . . . 5 ((𝑑:ω–1-1→𝑉 ∧ π‘Ž ∈ Ο‰) β†’ ((π‘ˆβ€˜π‘Ž) = ∩ ran π‘ˆ β†’ ∩ ran π‘ˆ β‰  βˆ…))
3534rexlimdva 3149 . . . 4 (𝑑:ω–1-1→𝑉 β†’ (βˆƒπ‘Ž ∈ Ο‰ (π‘ˆβ€˜π‘Ž) = ∩ ran π‘ˆ β†’ ∩ ran π‘ˆ β‰  βˆ…))
366, 35biimtrid 241 . . 3 (𝑑:ω–1-1→𝑉 β†’ (∩ ran π‘ˆ ∈ ran π‘ˆ β†’ ∩ ran π‘ˆ β‰  βˆ…))
3736adantl 481 . 2 ((βˆͺ ran 𝑑 ∈ 𝐹 ∧ 𝑑:ω–1-1→𝑉) β†’ (∩ ran π‘ˆ ∈ ran π‘ˆ β†’ ∩ ran π‘ˆ β‰  βˆ…))
383, 37mpd 15 1 ((βˆͺ ran 𝑑 ∈ 𝐹 ∧ 𝑑:ω–1-1→𝑉) β†’ ∩ ran π‘ˆ β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  {cab 2703   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  Vcvv 3468   βˆ– cdif 3940   ∩ cin 3942   βŠ† wss 3943  βˆ…c0 4317  ifcif 4523  π’« cpw 4597  {csn 4623  βˆͺ cuni 4902  βˆ© cint 4943   class class class wbr 5141  ran crn 5670  suc csuc 6360   Fn wfn 6532  β€“1-1β†’wf1 6534  β€“1-1-ontoβ†’wf1o 6536  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  Ο‰com 7852  seqΟ‰cseqom 8448   ↑m cmap 8822   β‰ˆ cen 8938  Fincfn 8941
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-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-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-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-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-seqom 8449  df-1o 8467  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945
This theorem is referenced by:  fin23lem31  10340
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