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Theorem fin23lem21 10283
Description: Lemma for fin23 10333. 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 10282 . 2 ((βˆͺ ran 𝑑 ∈ 𝐹 ∧ 𝑑:ω–1-1→𝑉) β†’ ∩ ran π‘ˆ ∈ ran π‘ˆ)
41fnseqom 8405 . . . . 5 π‘ˆ Fn Ο‰
5 fvelrnb 6907 . . . . 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 3451 . . . . . . . . . 10 𝑑 ∈ V
9 f1f1orn 6799 . . . . . . . . . 10 (𝑑:ω–1-1→𝑉 β†’ 𝑑:ω–1-1-ontoβ†’ran 𝑑)
10 f1oen3g 8912 . . . . . . . . . 10 ((𝑑 ∈ V ∧ 𝑑:ω–1-1-ontoβ†’ran 𝑑) β†’ Ο‰ β‰ˆ ran 𝑑)
118, 9, 10sylancr 588 . . . . . . . . 9 (𝑑:ω–1-1→𝑉 β†’ Ο‰ β‰ˆ ran 𝑑)
12 ominf 9208 . . . . . . . . 9 Β¬ Ο‰ ∈ Fin
13 ssdif0 4327 . . . . . . . . . . 11 (ran 𝑑 βŠ† {βˆ…} ↔ (ran 𝑑 βˆ– {βˆ…}) = βˆ…)
14 snfi 8994 . . . . . . . . . . . . 13 {βˆ…} ∈ Fin
15 ssfi 9123 . . . . . . . . . . . . 13 (({βˆ…} ∈ Fin ∧ ran 𝑑 βŠ† {βˆ…}) β†’ ran 𝑑 ∈ Fin)
1614, 15mpan 689 . . . . . . . . . . . 12 (ran 𝑑 βŠ† {βˆ…} β†’ ran 𝑑 ∈ Fin)
17 enfi 9140 . . . . . . . . . . . 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 4310 . . . . . . . . 9 ((ran 𝑑 βˆ– {βˆ…}) β‰  βˆ… ↔ βˆƒπ‘Ž π‘Ž ∈ (ran 𝑑 βˆ– {βˆ…}))
23 eldifsn 4751 . . . . . . . . . . 11 (π‘Ž ∈ (ran 𝑑 βˆ– {βˆ…}) ↔ (π‘Ž ∈ ran 𝑑 ∧ π‘Ž β‰  βˆ…))
24 elssuni 4902 . . . . . . . . . . . 12 (π‘Ž ∈ ran 𝑑 β†’ π‘Ž βŠ† βˆͺ ran 𝑑)
25 ssn0 4364 . . . . . . . . . . . 12 ((π‘Ž βŠ† βˆͺ ran 𝑑 ∧ π‘Ž β‰  βˆ…) β†’ βˆͺ ran 𝑑 β‰  βˆ…)
2624, 25sylan 581 . . . . . . . . . . 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 1934 . . . . . . . . 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 10277 . . . . . . 7 ((π‘Ž ∈ Ο‰ ∧ βˆͺ ran 𝑑 β‰  βˆ…) β†’ (π‘ˆβ€˜π‘Ž) β‰  βˆ…)
327, 30, 31syl2anr 598 . . . . . 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 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 483 . 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 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3447   βˆ– cdif 3911   ∩ cin 3913   βŠ† wss 3914  βˆ…c0 4286  ifcif 4490  π’« cpw 4564  {csn 4590  βˆͺ cuni 4869  βˆ© cint 4911   class class class wbr 5109  ran crn 5638  suc csuc 6323   Fn wfn 6495  β€“1-1β†’wf1 6497  β€“1-1-ontoβ†’wf1o 6499  β€˜cfv 6500  (class class class)co 7361   ∈ cmpo 7363  Ο‰com 7806  seqΟ‰cseqom 8397   ↑m cmap 8771   β‰ˆ cen 8886  Fincfn 8889
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-seqom 8398  df-1o 8416  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893
This theorem is referenced by:  fin23lem31  10287
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