Theorem smoiso 8335

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 7309	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
2		f1of 6808	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	1, 2	syl 17	. . 3 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴⟶𝐵)
4		ffdm 6723	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
5	4	simpld 498	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:dom 𝐹⟶𝐵)
6		fss 6710	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
7	5, 6	sylan 589	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
8	7	3adant2 1145	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
9	3, 8	syl3an1 1177	. 2 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
10		fdm 6703	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
11	10	eqcomd 2770	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
12		ordeq 6355	. . . . 5 ⊢ (𝐴 = dom 𝐹 → (Ord 𝐴 ↔ Ord dom 𝐹))
13	1, 2, 11, 12	4syl 19	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → (Ord 𝐴 ↔ Ord dom 𝐹))
14	13	biimpa 480	. . 3 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴) → Ord dom 𝐹)
15	14	3adant3 1146	. 2 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Ord dom 𝐹)
16	10	eleq2d 2850	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
17	10	eleq2d 2850	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
18	16, 17	anbi12d 641	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
19	1, 2, 18	3syl 18	. . . . 5 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
20		isorel 7312	. . . . . . . 8 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
21		epel 5552	. . . . . . . 8 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
22		fvex 6882	. . . . . . . . 9 ⊢ (𝐹‘𝑦) ∈ V
23	22	epeli 5551	. . . . . . . 8 ⊢ ((𝐹‘𝑥) E (𝐹‘𝑦) ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦))
24	20, 21, 23	3bitr3g 315	. . . . . . 7 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
25	24	biimpd 231	. . . . . 6 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
26	25	ex 416	. . . . 5 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦))))
27	19, 26	sylbid 242	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦))))
28	27	ralrimivv 3205	. . 3 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹(𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
29	28	3ad2ant1 1147	. 2 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹(𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
30		df-smo 8319	. 2 ⊢ (Smo 𝐹 ↔ (𝐹:dom 𝐹⟶On ∧ Ord dom 𝐹 ∧ ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹(𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦))))
31	9, 15, 29, 30	syl3anbrc 1358	1 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Smo 𝐹)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  ∀wral 3078   ⊆ wss 3906   class class class wbr 5102   E cep 5548  dom cdm 5649  Ord word 6347  Oncon0 6348  ⟶wf 6519  –1-1-onto→wf1o 6522  ‘cfv 6523   Isom wiso 6524  Smo wsmo 8318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-ord 6351  df-iota 6479  df-fn 6526  df-f 6527  df-f1 6528  df-f1o 6530  df-fv 6531  df-isom 6532  df-smo 8319

This theorem is referenced by:  smoiso2  8342