Theorem smoiso 8383

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 7330	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
2		f1of 6838	. . . 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 6753	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
5	4	simpld 493	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:dom 𝐹⟶𝐵)
6		fss 6739	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
7	5, 6	sylan 578	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
8	7	3adant2 1128	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
9	3, 8	syl3an1 1160	. 2 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
10		fdm 6732	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
11	10	eqcomd 2731	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
12		ordeq 6378	. . . . 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 475	. . 3 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴) → Ord dom 𝐹)
15	14	3adant3 1129	. 2 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Ord dom 𝐹)
16	10	eleq2d 2811	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
17	10	eleq2d 2811	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
18	16, 17	anbi12d 630	. . . . . 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 7333	. . . . . . . 8 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
21		epel 5585	. . . . . . . 8 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
22		fvex 6909	. . . . . . . . 9 ⊢ (𝐹‘𝑦) ∈ V
23	22	epeli 5584	. . . . . . . 8 ⊢ ((𝐹‘𝑥) E (𝐹‘𝑦) ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦))
24	20, 21, 23	3bitr3g 312	. . . . . . 7 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
25	24	biimpd 228	. . . . . 6 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
26	25	ex 411	. . . . 5 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦))))
27	19, 26	sylbid 239	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦))))
28	27	ralrimivv 3188	. . 3 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹(𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
29	28	3ad2ant1 1130	. 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 8367	. 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 1340	1 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Smo 𝐹)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  ∀wral 3050   ⊆ wss 3944   class class class wbr 5149   E cep 5581  dom cdm 5678  Ord word 6370  Oncon0 6371  ⟶wf 6545  –1-1-onto→wf1o 6548  ‘cfv 6549   Isom wiso 6550  Smo wsmo 8366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374  df-iota 6501  df-fn 6552  df-f 6553  df-f1 6554  df-f1o 6556  df-fv 6557  df-isom 6558  df-smo 8367

This theorem is referenced by:  smoiso2  8390