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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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