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Theorem fin23lem21 10333
Description: Lemma for fin23 10383. 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 10332 . 2 ((βˆͺ ran 𝑑 ∈ 𝐹 ∧ 𝑑:ω–1-1→𝑉) β†’ ∩ ran π‘ˆ ∈ ran π‘ˆ)
41fnseqom 8454 . . . . 5 π‘ˆ Fn Ο‰
5 fvelrnb 6952 . . . . 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 3478 . . . . . . . . . 10 𝑑 ∈ V
9 f1f1orn 6844 . . . . . . . . . 10 (𝑑:ω–1-1→𝑉 β†’ 𝑑:ω–1-1-ontoβ†’ran 𝑑)
10 f1oen3g 8961 . . . . . . . . . 10 ((𝑑 ∈ V ∧ 𝑑:ω–1-1-ontoβ†’ran 𝑑) β†’ Ο‰ β‰ˆ ran 𝑑)
118, 9, 10sylancr 587 . . . . . . . . 9 (𝑑:ω–1-1→𝑉 β†’ Ο‰ β‰ˆ ran 𝑑)
12 ominf 9257 . . . . . . . . 9 Β¬ Ο‰ ∈ Fin
13 ssdif0 4363 . . . . . . . . . . 11 (ran 𝑑 βŠ† {βˆ…} ↔ (ran 𝑑 βˆ– {βˆ…}) = βˆ…)
14 snfi 9043 . . . . . . . . . . . . 13 {βˆ…} ∈ Fin
15 ssfi 9172 . . . . . . . . . . . . 13 (({βˆ…} ∈ Fin ∧ ran 𝑑 βŠ† {βˆ…}) β†’ ran 𝑑 ∈ Fin)
1614, 15mpan 688 . . . . . . . . . . . 12 (ran 𝑑 βŠ† {βˆ…} β†’ ran 𝑑 ∈ Fin)
17 enfi 9189 . . . . . . . . . . . 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 2954 . . . . . . . . 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 4346 . . . . . . . . 9 ((ran 𝑑 βˆ– {βˆ…}) β‰  βˆ… ↔ βˆƒπ‘Ž π‘Ž ∈ (ran 𝑑 βˆ– {βˆ…}))
23 eldifsn 4790 . . . . . . . . . . 11 (π‘Ž ∈ (ran 𝑑 βˆ– {βˆ…}) ↔ (π‘Ž ∈ ran 𝑑 ∧ π‘Ž β‰  βˆ…))
24 elssuni 4941 . . . . . . . . . . . 12 (π‘Ž ∈ ran 𝑑 β†’ π‘Ž βŠ† βˆͺ ran 𝑑)
25 ssn0 4400 . . . . . . . . . . . 12 ((π‘Ž βŠ† βˆͺ ran 𝑑 ∧ π‘Ž β‰  βˆ…) β†’ βˆͺ ran 𝑑 β‰  βˆ…)
2624, 25sylan 580 . . . . . . . . . . 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 1933 . . . . . . . . 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 10327 . . . . . . 7 ((π‘Ž ∈ Ο‰ ∧ βˆͺ ran 𝑑 β‰  βˆ…) β†’ (π‘ˆβ€˜π‘Ž) β‰  βˆ…)
327, 30, 31syl2anr 597 . . . . . 6 ((𝑑:ω–1-1→𝑉 ∧ π‘Ž ∈ Ο‰) β†’ (π‘ˆβ€˜π‘Ž) β‰  βˆ…)
33 neeq1 3003 . . . . . 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 3155 . . . 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 482 . 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 396   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βˆ– cdif 3945   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  ifcif 4528  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908  βˆ© cint 4950   class class class wbr 5148  ran crn 5677  suc csuc 6366   Fn wfn 6538  β€“1-1β†’wf1 6540  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7408   ∈ cmpo 7410  Ο‰com 7854  seqΟ‰cseqom 8446   ↑m cmap 8819   β‰ˆ cen 8935  Fincfn 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-seqom 8447  df-1o 8465  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942
This theorem is referenced by:  fin23lem31  10337
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