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Theorem finnisoeu 10108
Description: A finite totally ordered set has a unique order isomorphism to a finite ordinal. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
finnisoeu ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ βˆƒ!𝑓 𝑓 Isom E , 𝑅 ((cardβ€˜π΄), 𝐴))
Distinct variable groups:   𝑅,𝑓   𝐴,𝑓

Proof of Theorem finnisoeu
StepHypRef Expression
1 eqid 2733 . . . . 5 OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐴)
21oiexg 9530 . . . 4 (𝐴 ∈ Fin β†’ OrdIso(𝑅, 𝐴) ∈ V)
32adantl 483 . . 3 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ OrdIso(𝑅, 𝐴) ∈ V)
4 simpr 486 . . . . 5 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ 𝐴 ∈ Fin)
5 wofi 9292 . . . . 5 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ 𝑅 We 𝐴)
61oiiso 9532 . . . . 5 ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) β†’ OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
74, 5, 6syl2anc 585 . . . 4 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
81oien 9533 . . . . . . . 8 ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) β†’ dom OrdIso(𝑅, 𝐴) β‰ˆ 𝐴)
94, 5, 8syl2anc 585 . . . . . . 7 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ dom OrdIso(𝑅, 𝐴) β‰ˆ 𝐴)
10 ficardid 9957 . . . . . . . . 9 (𝐴 ∈ Fin β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
1110adantl 483 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ (cardβ€˜π΄) β‰ˆ 𝐴)
1211ensymd 9001 . . . . . . 7 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ 𝐴 β‰ˆ (cardβ€˜π΄))
13 entr 9002 . . . . . . 7 ((dom OrdIso(𝑅, 𝐴) β‰ˆ 𝐴 ∧ 𝐴 β‰ˆ (cardβ€˜π΄)) β†’ dom OrdIso(𝑅, 𝐴) β‰ˆ (cardβ€˜π΄))
149, 12, 13syl2anc 585 . . . . . 6 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ dom OrdIso(𝑅, 𝐴) β‰ˆ (cardβ€˜π΄))
151oion 9531 . . . . . . . 8 (𝐴 ∈ Fin β†’ dom OrdIso(𝑅, 𝐴) ∈ On)
1615adantl 483 . . . . . . 7 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ dom OrdIso(𝑅, 𝐴) ∈ On)
17 ficardom 9956 . . . . . . . 8 (𝐴 ∈ Fin β†’ (cardβ€˜π΄) ∈ Ο‰)
1817adantl 483 . . . . . . 7 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ (cardβ€˜π΄) ∈ Ο‰)
19 onomeneq 9228 . . . . . . 7 ((dom OrdIso(𝑅, 𝐴) ∈ On ∧ (cardβ€˜π΄) ∈ Ο‰) β†’ (dom OrdIso(𝑅, 𝐴) β‰ˆ (cardβ€˜π΄) ↔ dom OrdIso(𝑅, 𝐴) = (cardβ€˜π΄)))
2016, 18, 19syl2anc 585 . . . . . 6 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ (dom OrdIso(𝑅, 𝐴) β‰ˆ (cardβ€˜π΄) ↔ dom OrdIso(𝑅, 𝐴) = (cardβ€˜π΄)))
2114, 20mpbid 231 . . . . 5 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ dom OrdIso(𝑅, 𝐴) = (cardβ€˜π΄))
22 isoeq4 7317 . . . . 5 (dom OrdIso(𝑅, 𝐴) = (cardβ€˜π΄) β†’ (OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((cardβ€˜π΄), 𝐴)))
2321, 22syl 17 . . . 4 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ (OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((cardβ€˜π΄), 𝐴)))
247, 23mpbid 231 . . 3 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((cardβ€˜π΄), 𝐴))
25 isoeq1 7314 . . 3 (𝑓 = OrdIso(𝑅, 𝐴) β†’ (𝑓 Isom E , 𝑅 ((cardβ€˜π΄), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((cardβ€˜π΄), 𝐴)))
263, 24, 25spcedv 3589 . 2 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ βˆƒπ‘“ 𝑓 Isom E , 𝑅 ((cardβ€˜π΄), 𝐴))
27 wemoiso2 7961 . . 3 (𝑅 We 𝐴 β†’ βˆƒ*𝑓 𝑓 Isom E , 𝑅 ((cardβ€˜π΄), 𝐴))
285, 27syl 17 . 2 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ βˆƒ*𝑓 𝑓 Isom E , 𝑅 ((cardβ€˜π΄), 𝐴))
29 df-eu 2564 . 2 (βˆƒ!𝑓 𝑓 Isom E , 𝑅 ((cardβ€˜π΄), 𝐴) ↔ (βˆƒπ‘“ 𝑓 Isom E , 𝑅 ((cardβ€˜π΄), 𝐴) ∧ βˆƒ*𝑓 𝑓 Isom E , 𝑅 ((cardβ€˜π΄), 𝐴)))
3026, 28, 29sylanbrc 584 1 ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) β†’ βˆƒ!𝑓 𝑓 Isom E , 𝑅 ((cardβ€˜π΄), 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆƒ*wmo 2533  βˆƒ!weu 2563  Vcvv 3475   class class class wbr 5149   E cep 5580   Or wor 5588   We wwe 5631  dom cdm 5677  Oncon0 6365  β€˜cfv 6544   Isom wiso 6545  Ο‰com 7855   β‰ˆ cen 8936  Fincfn 8939  OrdIsocoi 9504  cardccrd 9930
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-rep 5286  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-rmo 3377  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-se 5633  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-isom 6553  df-riota 7365  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-card 9934
This theorem is referenced by:  iunfictbso  10109
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