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Theorem finnisoeu 9147
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 2761 . . . . 5 OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐴)
21oiexg 8608 . . . 4 (𝐴 ∈ Fin → OrdIso(𝑅, 𝐴) ∈ V)
32adantl 473 . . 3 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) ∈ V)
4 simpr 479 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝐴 ∈ Fin)
5 wofi 8377 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝑅 We 𝐴)
61oiiso 8610 . . . . 5 ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
74, 5, 6syl2anc 696 . . . 4 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
81oien 8611 . . . . . . . 8 ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
94, 5, 8syl2anc 696 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
10 ficardid 8999 . . . . . . . . 9 (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
1110adantl 473 . . . . . . . 8 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (card‘𝐴) ≈ 𝐴)
1211ensymd 8175 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝐴 ≈ (card‘𝐴))
13 entr 8176 . . . . . . 7 ((dom OrdIso(𝑅, 𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴)) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
149, 12, 13syl2anc 696 . . . . . 6 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
151oion 8609 . . . . . . . 8 (𝐴 ∈ Fin → dom OrdIso(𝑅, 𝐴) ∈ On)
1615adantl 473 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ∈ On)
17 ficardom 8998 . . . . . . . 8 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
1817adantl 473 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (card‘𝐴) ∈ ω)
19 onomeneq 8318 . . . . . . 7 ((dom OrdIso(𝑅, 𝐴) ∈ On ∧ (card‘𝐴) ∈ ω) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
2016, 18, 19syl2anc 696 . . . . . 6 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
2114, 20mpbid 222 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) = (card‘𝐴))
22 isoeq4 6735 . . . . 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 222 . . 3 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴))
25 isoeq1 6732 . . . 4 (𝑓 = OrdIso(𝑅, 𝐴) → (𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴)))
2625spcegv 3435 . . 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 7321 . . 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 eu5 2634 . 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 701 1 ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wex 1853  wcel 2140  ∃!weu 2608  ∃*wmo 2609  Vcvv 3341   class class class wbr 4805   E cep 5179   Or wor 5187   We wwe 5225  dom cdm 5267  Oncon0 5885  cfv 6050   Isom wiso 6051  ωcom 7232  cen 8121  Fincfn 8124  OrdIsocoi 8582  cardccrd 8972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-se 5227  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-isom 6059  df-riota 6776  df-om 7233  df-wrecs 7578  df-recs 7639  df-1o 7731  df-er 7914  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-oi 8583  df-card 8976
This theorem is referenced by:  iunfictbso  9148
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