Theorem finnisoeu 10066

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 2729	. . . . 5 ⊢ OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐴)
2	1	oiexg 9488	. . . 4 ⊢ (𝐴 ∈ Fin → OrdIso(𝑅, 𝐴) ∈ V)
3	2	adantl 481	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) ∈ V)
4		simpr 484	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
5		wofi 9236	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝑅 We 𝐴)
6	1	oiiso 9490	. . . . 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 9491	. . . . . . . 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 9915	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
11	10	adantl 481	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (card‘𝐴) ≈ 𝐴)
12	11	ensymd 8976	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ≈ (card‘𝐴))
13		entr 8977	. . . . . . 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 9489	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → dom OrdIso(𝑅, 𝐴) ∈ On)
16	15	adantl 481	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ∈ On)
17		ficardom 9914	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
18	17	adantl 481	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (card‘𝐴) ∈ ω)
19		onomeneq 9178	. . . . . . 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 232	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) = (card‘𝐴))
22		isoeq4 7295	. . . . 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 232	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴))
25		isoeq1 7292	. . 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 3564	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
27		wemoiso2 7953	. . 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 2562	. 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 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  ∃!weu 2561  Vcvv 3447   class class class wbr 5107   E cep 5537   Or wor 5545   We wwe 5590  dom cdm 5638  Oncon0 6332  ‘cfv 6511   Isom wiso 6512  ωcom 7842   ≈ cen 8915  Fincfn 8918  OrdIsocoi 9462  cardccrd 9888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-om 7843  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-1o 8434  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-oi 9463  df-card 9892

This theorem is referenced by:  iunfictbso  10067