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Theorem finnisoeu 9531
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 2819 . . . . 5 OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐴)
21oiexg 8991 . . . 4 (𝐴 ∈ Fin → OrdIso(𝑅, 𝐴) ∈ V)
32adantl 484 . . 3 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) ∈ V)
4 simpr 487 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝐴 ∈ Fin)
5 wofi 8759 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝑅 We 𝐴)
61oiiso 8993 . . . . 5 ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
74, 5, 6syl2anc 586 . . . 4 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
81oien 8994 . . . . . . . 8 ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
94, 5, 8syl2anc 586 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
10 ficardid 9383 . . . . . . . . 9 (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
1110adantl 484 . . . . . . . 8 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (card‘𝐴) ≈ 𝐴)
1211ensymd 8552 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → 𝐴 ≈ (card‘𝐴))
13 entr 8553 . . . . . . 7 ((dom OrdIso(𝑅, 𝐴) ≈ 𝐴𝐴 ≈ (card‘𝐴)) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
149, 12, 13syl2anc 586 . . . . . 6 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
151oion 8992 . . . . . . . 8 (𝐴 ∈ Fin → dom OrdIso(𝑅, 𝐴) ∈ On)
1615adantl 484 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ∈ On)
17 ficardom 9382 . . . . . . . 8 (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
1817adantl 484 . . . . . . 7 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (card‘𝐴) ∈ ω)
19 onomeneq 8700 . . . . . . 7 ((dom OrdIso(𝑅, 𝐴) ∈ On ∧ (card‘𝐴) ∈ ω) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
2016, 18, 19syl2anc 586 . . . . . 6 ((𝑅 Or 𝐴𝐴 ∈ Fin) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
2114, 20mpbid 234 . . . . 5 ((𝑅 Or 𝐴𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) = (card‘𝐴))
22 isoeq4 7065 . . . . 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 234 . . 3 ((𝑅 Or 𝐴𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴))
25 isoeq1 7062 . . 3 (𝑓 = OrdIso(𝑅, 𝐴) → (𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴)))
263, 24, 25spcedv 3597 . 2 ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
27 wemoiso2 7667 . . 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 2648 . 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 585 1 ((𝑅 Or 𝐴𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1530  wex 1773  wcel 2107  ∃*wmo 2614  ∃!weu 2647  Vcvv 3493   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 7572  cen 8498  Fincfn 8501  OrdIsocoi 8965  cardccrd 9356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  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 7106  df-om 7573  df-wrecs 7939  df-recs 8000  df-1o 8094  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-oi 8966  df-card 9360
This theorem is referenced by:  iunfictbso  9532
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