Theorem finnisoeu 10107

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 2732	. . . . 5 ⊢ OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐴)
2	1	oiexg 9529	. . . 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 9291	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝑅 We 𝐴)
6	1	oiiso 9531	. . . . 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 9532	. . . . . . . 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 9956	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
11	10	adantl 482	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (card‘𝐴) ≈ 𝐴)
12	11	ensymd 9000	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ≈ (card‘𝐴))
13		entr 9001	. . . . . . 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 9530	. . . . . . . 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 9955	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
18	17	adantl 482	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (card‘𝐴) ∈ ω)
19		onomeneq 9227	. . . . . . 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 7316	. . . . 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 7313	. . 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 3588	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
27		wemoiso2 7960	. . 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 2563	. 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 1541  ∃wex 1781   ∈ wcel 2106  ∃*wmo 2532  ∃!weu 2562  Vcvv 3474   class class class wbr 5148   E cep 5579   Or wor 5587   We wwe 5630  dom cdm 5676  Oncon0 6364  ‘cfv 6543   Isom wiso 6544  ωcom 7854   ≈ cen 8935  Fincfn 8938  OrdIsocoi 9503  cardccrd 9929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-om 7855  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-1o 8465  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-oi 9504  df-card 9933

This theorem is referenced by:  iunfictbso  10108