Theorem smoiso 5835

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 5368	. . . 4 ⊢ (𝐹 Isom E , E (A, B) → 𝐹:A–1-1-onto→B)
2		f1of 5047	. . . 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 4982	. . . . . 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 4976	. . . . 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 909	. . 3 ⊢ ((𝐹:A⟶B ∧ Ord A ∧ B ⊆ On) → 𝐹:dom 𝐹⟶On)
9	3, 8	syl3an1 1152	. 2 ⊢ ((𝐹 Isom E , E (A, B) ∧ Ord A ∧ B ⊆ On) → 𝐹:dom 𝐹⟶On)
10		fdm 4972	. . . . . 6 ⊢ (𝐹:A⟶B → dom 𝐹 = A)
11	10	eqcomd 2023	. . . . 5 ⊢ (𝐹:A⟶B → A = dom 𝐹)
12		ordeq 4054	. . . . 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 910	. 2 ⊢ ((𝐹 Isom E , E (A, B) ∧ Ord A ∧ B ⊆ On) → Ord dom 𝐹)
16	10	eleq2d 2085	. . . . . . 7 ⊢ (𝐹:A⟶B → (x ∈ dom 𝐹 ↔ x ∈ A))
17	10	eleq2d 2085	. . . . . . 7 ⊢ (𝐹:A⟶B → (y ∈ dom 𝐹 ↔ y ∈ A))
18	16, 17	anbi12d 445	. . . . . 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 3999	. . . . . . . . 9 ⊢ (x E y ↔ x ∈ y)
21		isorel 5369	. . . . . . . . 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 4968	. . . . . . . . . . 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 472	. . . . . . . . 9 ⊢ ((𝐹 Isom E , E (A, B) ∧ (x ∈ A ∧ y ∈ A)) → y ∈ A)
27		funfvex 5113	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ y ∈ dom 𝐹) → (𝐹‘y) ∈ V)
28	27	funfni 4921	. . . . . . . . . 10 ⊢ ((𝐹 Fn A ∧ y ∈ A) → (𝐹‘y) ∈ V)
29		epelg 3997	. . . . . . . . . 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 393	. . . . . . . 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 2374	. . 3 ⊢ (𝐹 Isom E , E (A, B) → ∀x ∈ dom 𝐹∀y ∈ dom 𝐹(x ∈ y → (𝐹‘x) ∈ (𝐹‘y)))
37	36	3ad2ant1 911	. 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 5819	. 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 1073	1 ⊢ ((𝐹 Isom E , E (A, B) ∧ Ord A ∧ B ⊆ On) → Smo 𝐹)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 871   = wceq 1226   ∈ wcel 1370  ∀wral 2280  Vcvv 2531   ⊆ wss 2890   class class class wbr 3734   E cep 3994  Ord word 4044  Oncon0 4045  dom cdm 4268   Fn wfn 4820  ⟶wf 4821  –1-1-onto→wf1o 4824  ‘cfv 4825   Isom wiso 4826  Smo wsmo 5818

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914

This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-tr 3825  df-eprel 3996  df-id 4000  df-iord 4048  df-cnv 4276  df-co 4277  df-dm 4278  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-f1o 4832  df-fv 4833  df-isom 4834  df-smo 5819

This theorem is referenced by: (None)