Theorem smoiso 8349

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 7322	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
2		f1of 6821	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵)
3	1, 2	syl 18	. . 3 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴⟶𝐵)
4		ffdm 6736	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
5	4	simpld 499	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:dom 𝐹⟶𝐵)
6		fss 6723	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
7	5, 6	sylan 591	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
8	7	3adant2 1147	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
9	3, 8	syl3an1 1179	. 2 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
10		fdm 6716	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
11	10	eqcomd 2775	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
12		ordeq 6368	. . . . 5 ⊢ (𝐴 = dom 𝐹 → (Ord 𝐴 ↔ Ord dom 𝐹))
13	1, 2, 11, 12	4syl 20	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → (Ord 𝐴 ↔ Ord dom 𝐹))
14	13	biimpa 481	. . 3 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴) → Ord dom 𝐹)
15	14	3adant3 1148	. 2 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Ord dom 𝐹)
16	10	eleq2d 2855	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
17	10	eleq2d 2855	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
18	16, 17	anbi12d 643	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
19	1, 2, 18	3syl 19	. . . . 5 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
20		isorel 7325	. . . . . . . 8 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
21		epel 5565	. . . . . . . 8 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
22		fvex 6895	. . . . . . . . 9 ⊢ (𝐹‘𝑦) ∈ V
23	22	epeli 5564	. . . . . . . 8 ⊢ ((𝐹‘𝑥) E (𝐹‘𝑦) ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦))
24	20, 21, 23	3bitr3g 316	. . . . . . 7 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
25	24	biimpd 232	. . . . . 6 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
26	25	ex 417	. . . . 5 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦))))
27	19, 26	sylbid 243	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦))))
28	27	ralrimivv 3212	. . 3 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹(𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
29	28	3ad2ant1 1149	. 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 8333	. 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 1360	1 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Smo 𝐹)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3911   class class class wbr 5111   E cep 5561  dom cdm 5662  Ord word 6360  Oncon0 6361  ⟶wf 6533  –1-1-onto→wf1o 6536  ‘cfv 6537   Isom wiso 6538  Smo wsmo 8332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5259  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-iota 6493  df-fn 6540  df-f 6541  df-f1 6542  df-f1o 6544  df-fv 6545  df-isom 6546  df-smo 8333

This theorem is referenced by:  smoiso2  8356