Theorem smoiso 6192

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 5701	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → 𝐹:𝐴–1-1-onto→𝐵)
2		f1of 5360	. . . 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 5288	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹:dom 𝐹⟶𝐵 ∧ dom 𝐹 ⊆ 𝐴))
5	4	simpld 111	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹:dom 𝐹⟶𝐵)
6		fss 5279	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
7	5, 6	sylan 281	. . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
8	7	3adant2 1000	. . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
9	3, 8	syl3an1 1249	. 2 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → 𝐹:dom 𝐹⟶On)
10		fdm 5273	. . . . . 6 ⊢ (𝐹:𝐴⟶𝐵 → dom 𝐹 = 𝐴)
11	10	eqcomd 2143	. . . . 5 ⊢ (𝐹:𝐴⟶𝐵 → 𝐴 = dom 𝐹)
12		ordeq 4289	. . . . 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 294	. . 3 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴) → Ord dom 𝐹)
15	14	3adant3 1001	. 2 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Ord dom 𝐹)
16	10	eleq2d 2207	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
17	10	eleq2d 2207	. . . . . . 7 ⊢ (𝐹:𝐴⟶𝐵 → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
18	16, 17	anbi12d 464	. . . . . 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 4209	. . . . . . . . 9 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
21		isorel 5702	. . . . . . . . 9 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
22	20, 21	syl5bbr 193	. . . . . . . 8 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ↔ (𝐹‘𝑥) E (𝐹‘𝑦)))
23		ffn 5267	. . . . . . . . . . 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 274	. . . . . . . . 9 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝐹 Fn 𝐴)
26		simprr 521	. . . . . . . . 9 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
27		funfvex 5431	. . . . . . . . . . 11 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ V)
28	27	funfni 5218	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝐹‘𝑦) ∈ V)
29		epelg 4207	. . . . . . . . . 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 408	. . . . . . . 8 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝐹‘𝑥) E (𝐹‘𝑦) ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
32	22, 31	bitrd 187	. . . . . . 7 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ↔ (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
33	32	biimpd 143	. . . . . 6 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
34	33	ex 114	. . . . 5 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦))))
35	19, 34	sylbid 149	. . . 4 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦))))
36	35	ralrimivv 2511	. . 3 ⊢ (𝐹 Isom E , E (𝐴, 𝐵) → ∀𝑥 ∈ dom 𝐹∀𝑦 ∈ dom 𝐹(𝑥 ∈ 𝑦 → (𝐹‘𝑥) ∈ (𝐹‘𝑦)))
37	36	3ad2ant1 1002	. 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 6176	. 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 1165	1 ⊢ ((𝐹 Isom E , E (𝐴, 𝐵) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On) → Smo 𝐹)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2414  Vcvv 2681   ⊆ wss 3066   class class class wbr 3924   E cep 4204  Ord word 4279  Oncon0 4280  dom cdm 4534   Fn wfn 5113  ⟶wf 5114  –1-1-onto→wf1o 5117  ‘cfv 5118   Isom wiso 5119  Smo wsmo 6175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-tr 4022  df-eprel 4206  df-id 4210  df-iord 4283  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-f1o 5125  df-fv 5126  df-isom 5127  df-smo 6176

This theorem is referenced by: (None)