Theorem finnisoeu 9869

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 𝐴 ∧ 𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))

Distinct variable groups:   𝑅,𝑓   𝐴,𝑓

Proof of Theorem finnisoeu

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . 5 ⊢ OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐴)
2	1	oiexg 9294	. . . 4 ⊢ (𝐴 ∈ Fin → OrdIso(𝑅, 𝐴) ∈ V)
3	2	adantl 482	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) ∈ V)
4		simpr 485	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
5		wofi 9063	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝑅 We 𝐴)
6	1	oiiso 9296	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
7	4, 5, 6	syl2anc 584	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
8	1	oien 9297	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
9	4, 5, 8	syl2anc 584	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
10		ficardid 9720	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
11	10	adantl 482	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (card‘𝐴) ≈ 𝐴)
12	11	ensymd 8791	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ≈ (card‘𝐴))
13		entr 8792	. . . . . . 7 ⊢ ((dom OrdIso(𝑅, 𝐴) ≈ 𝐴 ∧ 𝐴 ≈ (card‘𝐴)) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
14	9, 12, 13	syl2anc 584	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
15	1	oion 9295	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → dom OrdIso(𝑅, 𝐴) ∈ On)
16	15	adantl 482	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ∈ On)
17		ficardom 9719	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
18	17	adantl 482	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (card‘𝐴) ∈ ω)
19		onomeneq 9011	. . . . . . 7 ⊢ ((dom OrdIso(𝑅, 𝐴) ∈ On ∧ (card‘𝐴) ∈ ω) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
20	16, 18, 19	syl2anc 584	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
21	14, 20	mpbid 231	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) = (card‘𝐴))
22		isoeq4 7191	. . . . 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)))
23	21, 22	syl 17	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴)))
24	7, 23	mpbid 231	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴))
25		isoeq1 7188	. . 3 ⊢ (𝑓 = OrdIso(𝑅, 𝐴) → (𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴)))
26	3, 24, 25	spcedv 3537	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
27		wemoiso2 7817	. . 3 ⊢ (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
28	5, 27	syl 17	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃*𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
29		df-eu 2569	. 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)))
30	26, 28, 29	sylanbrc 583	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ∃*wmo 2538  ∃!weu 2568  Vcvv 3432   class class class wbr 5074   E cep 5494   Or wor 5502   We wwe 5543  dom cdm 5589  Oncon0 6266  ‘cfv 6433   Isom wiso 6434  ωcom 7712   ≈ cen 8730  Fincfn 8733  OrdIsocoi 9268  cardccrd 9693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-oi 9269  df-card 9697

This theorem is referenced by:  iunfictbso  9870