Theorem finnisoeu 8936

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 2622	. . . . 5 ⊢ OrdIso(𝑅, 𝐴) = OrdIso(𝑅, 𝐴)
2	1	oiexg 8440	. . . 4 ⊢ (𝐴 ∈ Fin → OrdIso(𝑅, 𝐴) ∈ V)
3	2	adantl 482	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) ∈ V)
4		simpr 477	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin)
5		wofi 8209	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝑅 We 𝐴)
6	1	oiiso 8442	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
7	4, 5, 6	syl2anc 693	. . . 4 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 (dom OrdIso(𝑅, 𝐴), 𝐴))
8	1	oien 8443	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝑅 We 𝐴) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
9	4, 5, 8	syl2anc 693	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ 𝐴)
10		ficardid 8788	. . . . . . . . 9 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ≈ 𝐴)
11	10	adantl 482	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (card‘𝐴) ≈ 𝐴)
12	11	ensymd 8007	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → 𝐴 ≈ (card‘𝐴))
13		entr 8008	. . . . . . 7 ⊢ ((dom OrdIso(𝑅, 𝐴) ≈ 𝐴 ∧ 𝐴 ≈ (card‘𝐴)) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
14	9, 12, 13	syl2anc 693	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴))
15	1	oion 8441	. . . . . . . 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 8787	. . . . . . . 8 ⊢ (𝐴 ∈ Fin → (card‘𝐴) ∈ ω)
18	17	adantl 482	. . . . . . 7 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (card‘𝐴) ∈ ω)
19		onomeneq 8150	. . . . . . 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 693	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → (dom OrdIso(𝑅, 𝐴) ≈ (card‘𝐴) ↔ dom OrdIso(𝑅, 𝐴) = (card‘𝐴)))
21	14, 20	mpbid 222	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → dom OrdIso(𝑅, 𝐴) = (card‘𝐴))
22		isoeq4 6570	. . . . 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 222	. . 3 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴))
25		isoeq1 6567	. . . 4 ⊢ (𝑓 = OrdIso(𝑅, 𝐴) → (𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴) ↔ OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴)))
26	25	spcegv 3294	. . 3 ⊢ (OrdIso(𝑅, 𝐴) ∈ V → (OrdIso(𝑅, 𝐴) Isom E , 𝑅 ((card‘𝐴), 𝐴) → ∃𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴)))
27	3, 24, 26	sylc 65	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
28		wemoiso2 7154	. . 3 ⊢ (𝑅 We 𝐴 → ∃*𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
29	5, 28	syl 17	. 2 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃*𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))
30		eu5 2496	. 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)))
31	27, 29, 30	sylanbrc 698	1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐴 ∈ Fin) → ∃!𝑓 𝑓 Isom E , 𝑅 ((card‘𝐴), 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  ∃*wmo 2471  Vcvv 3200   class class class wbr 4653   E cep 5028   Or wor 5034   We wwe 5072  dom cdm 5114  Oncon0 5723  ‘cfv 5888   Isom wiso 5889  ωcom 7065   ≈ cen 7952  Fincfn 7955  OrdIsocoi 8414  cardccrd 8761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-om 7066  df-wrecs 7407  df-recs 7468  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765

This theorem is referenced by:  iunfictbso  8937