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Theorem finnisoeu 9222
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 2799 . . . . 5 OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐴)
21oiexg 8682 . . . 4 (𝐴 ∈ Fin → OrdIso(𝑅, 𝐴) ∈ V)
32adantl 474 . . 3 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) ∈ V)
4 simpr 478 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝐴 ∈ Fin)
5 wofi 8451 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝑅 We 𝐴)
61oiiso 8684 . . . . 5 ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
74, 5, 6syl2anc 580 . . . 4 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
81oien 8685 . . . . . . . 8 ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
94, 5, 8syl2anc 580 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
10 ficardid 9074 . . . . . . . . 9 (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
1110adantl 474 . . . . . . . 8 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (card‘𝐴) ≈ 𝐴)
1211ensymd 8246 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝐴 ≈ (card‘𝐴))
13 entr 8247 . . . . . . 7 ((dom OrdIso(𝑅, 𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴)) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
149, 12, 13syl2anc 580 . . . . . 6 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
151oion 8683 . . . . . . . 8 (𝐴 ∈ Fin → dom OrdIso(𝑅, 𝐴) ∈ On)
1615adantl 474 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ∈ On)
17 ficardom 9073 . . . . . . . 8 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
1817adantl 474 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (card‘𝐴) ∈ ω)
19 onomeneq 8392 . . . . . . 7 ((dom OrdIso(𝑅, 𝐴) ∈ On ∧ (card‘𝐴) ∈ ω) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
2016, 18, 19syl2anc 580 . . . . . 6 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
2114, 20mpbid 224 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) = (card‘𝐴))
22 isoeq4 6798 . . . . 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 224 . . 3 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴))
25 isoeq1 6795 . . . 4 (𝑓 = OrdIso(𝑅, 𝐴) → (𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴)))
2625spcegv 3482 . . 3 (OrdIso(𝑅, 𝐴) ∈ V → (OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴) → ∃𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴)))
273, 24, 26sylc 65 . 2 ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
28 wemoiso2 7387 . . 3 (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
295, 28syl 17 . 2 ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃*𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
30 df-eu 2609 . 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)))
3127, 29, 30sylanbrc 579 1 ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wex 1875  wcel 2157  ∃*wmo 2589  ∃!weu 2608  Vcvv 3385   class class class wbr 4843   E cep 5224   Or wor 5232   We wwe 5270  dom cdm 5312  Oncon0 5941  cfv 6101   Isom wiso 6102  ωcom 7299  cen 8192  Fincfn 8195  OrdIsocoi 8656  cardccrd 9047
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-om 7300  df-wrecs 7645  df-recs 7707  df-1o 7799  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-oi 8657  df-card 9051
This theorem is referenced by:  iunfictbso  9223
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