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Theorem finnisoeu 9527
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 𝐴𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
Distinct variable groups:   𝑅,𝑓   𝐴,𝑓

Proof of Theorem finnisoeu
StepHypRef Expression
1 eqid 2818 . . . . 5 OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐴)
21oiexg 8987 . . . 4 (𝐴 ∈ Fin → OrdIso(𝑅, 𝐴) ∈ V)
32adantl 482 . . 3 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) ∈ V)
4 simpr 485 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝐴 ∈ Fin)
5 wofi 8755 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝑅 We 𝐴)
61oiiso 8989 . . . . 5 ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
74, 5, 6syl2anc 584 . . . 4 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
81oien 8990 . . . . . . . 8 ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
94, 5, 8syl2anc 584 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
10 ficardid 9379 . . . . . . . . 9 (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
1110adantl 482 . . . . . . . 8 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (card‘𝐴) ≈ 𝐴)
1211ensymd 8548 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝐴 ≈ (card‘𝐴))
13 entr 8549 . . . . . . 7 ((dom OrdIso(𝑅, 𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴)) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
149, 12, 13syl2anc 584 . . . . . 6 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
151oion 8988 . . . . . . . 8 (𝐴 ∈ Fin → dom OrdIso(𝑅, 𝐴) ∈ On)
1615adantl 482 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ∈ On)
17 ficardom 9378 . . . . . . . 8 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
1817adantl 482 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (card‘𝐴) ∈ ω)
19 onomeneq 8696 . . . . . . 7 ((dom OrdIso(𝑅, 𝐴) ∈ On ∧ (card‘𝐴) ∈ ω) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
2016, 18, 19syl2anc 584 . . . . . 6 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
2114, 20mpbid 233 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) = (card‘𝐴))
22 isoeq4 7062 . . . . 5 (dom OrdIso(𝑅, 𝐴) = (card‘𝐴) → (OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴)))
2321, 22syl 17 . . . 4 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴)))
247, 23mpbid 233 . . 3 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴))
25 isoeq1 7059 . . 3 (𝑓 = OrdIso(𝑅, 𝐴) → (𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴)))
263, 24, 25spcedv 3596 . 2 ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
27 wemoiso2 7664 . . 3 (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
285, 27syl 17 . 2 ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃*𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
29 df-eu 2647 . 2 (∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴) ↔ (∃𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴) ∧ ∃*𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴)))
3026, 28, 29sylanbrc 583 1 ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  ∃*wmo 2613  ∃!weu 2646  Vcvv 3492   class class class wbr 5057   E cep 5457   Or wor 5466   We wwe 5506  dom cdm 5548  Oncon0 6184  cfv 6348   Isom wiso 6349  ωcom 7569  cen 8494  Fincfn 8497  OrdIsocoi 8961  cardccrd 9352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-om 7570  df-wrecs 7936  df-recs 7997  df-1o 8091  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-card 9356
This theorem is referenced by:  iunfictbso  9528
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