Metamath Proof Explorer


Theorem smoiso

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)

Ref Expression
Assertion smoiso ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴𝐵 ⊆ On ) → Smo 𝐹 )

Proof

Step Hyp Ref Expression
1 isof1o ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → 𝐹 : 𝐴1-1-onto𝐵 )
2 f1of ( 𝐹 : 𝐴1-1-onto𝐵𝐹 : 𝐴𝐵 )
3 1 2 syl ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → 𝐹 : 𝐴𝐵 )
4 ffdm ( 𝐹 : 𝐴𝐵 → ( 𝐹 : dom 𝐹𝐵 ∧ dom 𝐹𝐴 ) )
5 4 simpld ( 𝐹 : 𝐴𝐵𝐹 : dom 𝐹𝐵 )
6 fss ( ( 𝐹 : dom 𝐹𝐵𝐵 ⊆ On ) → 𝐹 : dom 𝐹 ⟶ On )
7 5 6 sylan ( ( 𝐹 : 𝐴𝐵𝐵 ⊆ On ) → 𝐹 : dom 𝐹 ⟶ On )
8 7 3adant2 ( ( 𝐹 : 𝐴𝐵 ∧ Ord 𝐴𝐵 ⊆ On ) → 𝐹 : dom 𝐹 ⟶ On )
9 3 8 syl3an1 ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴𝐵 ⊆ On ) → 𝐹 : dom 𝐹 ⟶ On )
10 fdm ( 𝐹 : 𝐴𝐵 → dom 𝐹 = 𝐴 )
11 10 eqcomd ( 𝐹 : 𝐴𝐵𝐴 = dom 𝐹 )
12 ordeq ( 𝐴 = dom 𝐹 → ( Ord 𝐴 ↔ Ord dom 𝐹 ) )
13 1 2 11 12 4syl ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → ( Ord 𝐴 ↔ Ord dom 𝐹 ) )
14 13 biimpa ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ) → Ord dom 𝐹 )
15 14 3adant3 ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴𝐵 ⊆ On ) → Ord dom 𝐹 )
16 10 eleq2d ( 𝐹 : 𝐴𝐵 → ( 𝑥 ∈ dom 𝐹𝑥𝐴 ) )
17 10 eleq2d ( 𝐹 : 𝐴𝐵 → ( 𝑦 ∈ dom 𝐹𝑦𝐴 ) )
18 16 17 anbi12d ( 𝐹 : 𝐴𝐵 → ( ( 𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹 ) ↔ ( 𝑥𝐴𝑦𝐴 ) ) )
19 1 2 18 3syl ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → ( ( 𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹 ) ↔ ( 𝑥𝐴𝑦𝐴 ) ) )
20 isorel ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( 𝑥 E 𝑦 ↔ ( 𝐹𝑥 ) E ( 𝐹𝑦 ) ) )
21 epel ( 𝑥 E 𝑦𝑥𝑦 )
22 fvex ( 𝐹𝑦 ) ∈ V
23 22 epeli ( ( 𝐹𝑥 ) E ( 𝐹𝑦 ) ↔ ( 𝐹𝑥 ) ∈ ( 𝐹𝑦 ) )
24 20 21 23 3bitr3g ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( 𝑥𝑦 ↔ ( 𝐹𝑥 ) ∈ ( 𝐹𝑦 ) ) )
25 24 biimpd ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( 𝑥𝑦 → ( 𝐹𝑥 ) ∈ ( 𝐹𝑦 ) ) )
26 25 ex ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → ( ( 𝑥𝐴𝑦𝐴 ) → ( 𝑥𝑦 → ( 𝐹𝑥 ) ∈ ( 𝐹𝑦 ) ) ) )
27 19 26 sylbid ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → ( ( 𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹 ) → ( 𝑥𝑦 → ( 𝐹𝑥 ) ∈ ( 𝐹𝑦 ) ) ) )
28 27 ralrimivv ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → ∀ 𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹 ( 𝑥𝑦 → ( 𝐹𝑥 ) ∈ ( 𝐹𝑦 ) ) )
29 28 3ad2ant1 ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴𝐵 ⊆ On ) → ∀ 𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹 ( 𝑥𝑦 → ( 𝐹𝑥 ) ∈ ( 𝐹𝑦 ) ) )
30 df-smo ( Smo 𝐹 ↔ ( 𝐹 : dom 𝐹 ⟶ On ∧ Ord dom 𝐹 ∧ ∀ 𝑥 ∈ dom 𝐹𝑦 ∈ dom 𝐹 ( 𝑥𝑦 → ( 𝐹𝑥 ) ∈ ( 𝐹𝑦 ) ) ) )
31 9 15 29 30 syl3anbrc ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴𝐵 ⊆ On ) → Smo 𝐹 )