Theorem smoiso 5858

Description: If 𝐹 is an isomorphism from an ordinal A onto B, which is a subset of the ordinals, then 𝐹 is a strictly monotonic function. Exercise 3 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 24-Nov-2011.)

Assertion

Ref	Expression
smoiso	⊢ ((𝐹 Isom E , E (A, B) ∧ Ord A ∧ B ⊆ On) → Smo 𝐹)

Proof of Theorem smoiso

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isof1o 5390	. . . 4 ⊢ (𝐹 Isom E , E (A, B) → 𝐹:A–1-1-onto→B)
2		f1of 5069	. . . 4 ⊢ (𝐹:A–1-1-onto→B → 𝐹:A⟶B)
3	1, 2	syl 14	. . 3 ⊢ (𝐹 Isom E , E (A, B) → 𝐹:A⟶B)
4		ffdm 5004	. . . . . 6 ⊢ (𝐹:A⟶B → (𝐹:dom 𝐹⟶B ∧ dom 𝐹 ⊆ A))
5	4	simpld 105	. . . . 5 ⊢ (𝐹:A⟶B → 𝐹:dom 𝐹⟶B)
6		fss 4997	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶B ∧ B ⊆ On) → 𝐹:dom 𝐹⟶On)
7	5, 6	sylan 267	. . . 4 ⊢ ((𝐹:A⟶B ∧ B ⊆ On) → 𝐹:dom 𝐹⟶On)
8	7	3adant2 922	. . 3 ⊢ ((𝐹:A⟶B ∧ Ord A ∧ B ⊆ On) → 𝐹:dom 𝐹⟶On)
9	3, 8	syl3an1 1167	. 2 ⊢ ((𝐹 Isom E , E (A, B) ∧ Ord A ∧ B ⊆ On) → 𝐹:dom 𝐹⟶On)
10		fdm 4993	. . . . . 6 ⊢ (𝐹:A⟶B → dom 𝐹 = A)
11	10	eqcomd 2042	. . . . 5 ⊢ (𝐹:A⟶B → A = dom 𝐹)
12		ordeq 4075	. . . . 5 ⊢ (A = dom 𝐹 → (Ord A ↔ Ord dom 𝐹))
13	1, 2, 11, 12	4syl 18	. . . 4 ⊢ (𝐹 Isom E , E (A, B) → (Ord A ↔ Ord dom 𝐹))
14	13	biimpa 280	. . 3 ⊢ ((𝐹 Isom E , E (A, B) ∧ Ord A) → Ord dom 𝐹)
15	14	3adant3 923	. 2 ⊢ ((𝐹 Isom E , E (A, B) ∧ Ord A ∧ B ⊆ On) → Ord dom 𝐹)
16	10	eleq2d 2104	. . . . . . 7 ⊢ (𝐹:A⟶B → (x ∈ dom 𝐹 ↔ x ∈ A))
17	10	eleq2d 2104	. . . . . . 7 ⊢ (𝐹:A⟶B → (y ∈ dom 𝐹 ↔ y ∈ A))
18	16, 17	anbi12d 442	. . . . . 6 ⊢ (𝐹:A⟶B → ((x ∈ dom 𝐹 ∧ y ∈ dom 𝐹) ↔ (x ∈ A ∧ y ∈ A)))
19	1, 2, 18	3syl 17	. . . . 5 ⊢ (𝐹 Isom E , E (A, B) → ((x ∈ dom 𝐹 ∧ y ∈ dom 𝐹) ↔ (x ∈ A ∧ y ∈ A)))
20		epel 4020	. . . . . . . . 9 ⊢ (x E y ↔ x ∈ y)
21		isorel 5391	. . . . . . . . 9 ⊢ ((𝐹 Isom E , E (A, B) ∧ (x ∈ A ∧ y ∈ A)) → (x E y ↔ (𝐹‘x) E (𝐹‘y)))
22	20, 21	syl5bbr 183	. . . . . . . 8 ⊢ ((𝐹 Isom E , E (A, B) ∧ (x ∈ A ∧ y ∈ A)) → (x ∈ y ↔ (𝐹‘x) E (𝐹‘y)))
23		ffn 4989	. . . . . . . . . . 11 ⊢ (𝐹:A⟶B → 𝐹 Fn A)
24	3, 23	syl 14	. . . . . . . . . 10 ⊢ (𝐹 Isom E , E (A, B) → 𝐹 Fn A)
25	24	adantr 261	. . . . . . . . 9 ⊢ ((𝐹 Isom E , E (A, B) ∧ (x ∈ A ∧ y ∈ A)) → 𝐹 Fn A)
26		simprr 484	. . . . . . . . 9 ⊢ ((𝐹 Isom E , E (A, B) ∧ (x ∈ A ∧ y ∈ A)) → y ∈ A)
27		funfvex 5135	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ y ∈ dom 𝐹) → (𝐹‘y) ∈ V)
28	27	funfni 4942	. . . . . . . . . 10 ⊢ ((𝐹 Fn A ∧ y ∈ A) → (𝐹‘y) ∈ V)
29		epelg 4018	. . . . . . . . . 10 ⊢ ((𝐹‘y) ∈ V → ((𝐹‘x) E (𝐹‘y) ↔ (𝐹‘x) ∈ (𝐹‘y)))
30	28, 29	syl 14	. . . . . . . . 9 ⊢ ((𝐹 Fn A ∧ y ∈ A) → ((𝐹‘x) E (𝐹‘y) ↔ (𝐹‘x) ∈ (𝐹‘y)))
31	25, 26, 30	syl2anc 391	. . . . . . . 8 ⊢ ((𝐹 Isom E , E (A, B) ∧ (x ∈ A ∧ y ∈ A)) → ((𝐹‘x) E (𝐹‘y) ↔ (𝐹‘x) ∈ (𝐹‘y)))
32	22, 31	bitrd 177	. . . . . . 7 ⊢ ((𝐹 Isom E , E (A, B) ∧ (x ∈ A ∧ y ∈ A)) → (x ∈ y ↔ (𝐹‘x) ∈ (𝐹‘y)))
33	32	biimpd 132	. . . . . 6 ⊢ ((𝐹 Isom E , E (A, B) ∧ (x ∈ A ∧ y ∈ A)) → (x ∈ y → (𝐹‘x) ∈ (𝐹‘y)))
34	33	ex 108	. . . . 5 ⊢ (𝐹 Isom E , E (A, B) → ((x ∈ A ∧ y ∈ A) → (x ∈ y → (𝐹‘x) ∈ (𝐹‘y))))
35	19, 34	sylbid 139	. . . 4 ⊢ (𝐹 Isom E , E (A, B) → ((x ∈ dom 𝐹 ∧ y ∈ dom 𝐹) → (x ∈ y → (𝐹‘x) ∈ (𝐹‘y))))
36	35	ralrimivv 2394	. . 3 ⊢ (𝐹 Isom E , E (A, B) → ∀x ∈ dom 𝐹∀y ∈ dom 𝐹(x ∈ y → (𝐹‘x) ∈ (𝐹‘y)))
37	36	3ad2ant1 924	. 2 ⊢ ((𝐹 Isom E , E (A, B) ∧ Ord A ∧ B ⊆ On) → ∀x ∈ dom 𝐹∀y ∈ dom 𝐹(x ∈ y → (𝐹‘x) ∈ (𝐹‘y)))
38		df-smo 5842	. 2 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀x ∈ dom 𝐹∀y ∈ dom 𝐹(x ∈ y → (𝐹‘x) ∈ (𝐹‘y))))
39	9, 15, 37, 38	syl3anbrc 1087	1 ⊢ ((𝐹 Isom E , E (A, B) ∧ Ord A ∧ B ⊆ On) → Smo 𝐹)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ⊆ wss 2911   class class class wbr 3755   E cep 4015  Ord word 4065  Oncon0 4066  dom cdm 4288   Fn wfn 4840  ⟶wf 4841  –1-1-onto→wf1o 4844  ‘cfv 4845   Isom wiso 4846  Smo wsmo 5841

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-tr 3846  df-eprel 4017  df-id 4021  df-iord 4069  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-f1o 4852  df-fv 4853  df-isom 4854  df-smo 5842

This theorem is referenced by: (None)