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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 𝐴 ∧ 𝐵 ⊆ On ) → Smo 𝐹 )

Proof

Step Hyp Ref Expression
1 isof1o ⊢ ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → 𝐹 : 𝐴 –1-1-onto→ 𝐵 )
2 f1of ⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )
3 1 2 syl ⊢ ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → 𝐹 : 𝐴 ⟶ 𝐵 )
4 ffdm ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝐹 : dom 𝐹 ⟶ 𝐵 ∧ dom 𝐹 ⊆ 𝐴 ) )
5 4 simpld ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 : dom 𝐹 ⟶ 𝐵 )
6 fss ⊢ ( ( 𝐹 : dom 𝐹 ⟶ 𝐵 ∧ 𝐵 ⊆ On ) → 𝐹 : dom 𝐹 ⟶ On )
7 5 6 sylan ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝐵 ⊆ On ) → 𝐹 : dom 𝐹 ⟶ On )
8 7 3adant2 ⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Ord 𝐴 ∧ 𝐵 ⊆ On ) → 𝐹 : dom 𝐹 ⟶ On )
9 3 8 syl3an1 ⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On ) → 𝐹 : dom 𝐹 ⟶ On )
10 fdm ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → dom 𝐹 = 𝐴 )
11 10 eqcomd ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐴 = 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 𝐴 ∧ 𝐵 ⊆ On ) → Ord dom 𝐹 )
16 10 eleq2d ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴 ) )
17 10 eleq2d ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( 𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴 ) )
18 16 17 anbi12d ⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ( ( 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) )
19 1 2 18 3syl ⊢ ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → ( ( 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) )
20 isorel ⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 E 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) E ( 𝐹 ‘ 𝑦 ) ) )
21 epel ⊢ ( 𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦 )
22 fvex ⊢ ( 𝐹 ‘ 𝑦 ) ∈ V
23 22 epeli ⊢ ( ( 𝐹 ‘ 𝑥 ) E ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑦 ) )
24 20 21 23 3bitr3g ⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 ∈ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑦 ) ) )
25 24 biimpd ⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) ) → ( 𝑥 ∈ 𝑦 → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑦 ) ) )
26 25 ex ⊢ ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ) → ( 𝑥 ∈ 𝑦 → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) )
27 19 26 sylbid ⊢ ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → ( ( 𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ dom 𝐹 ) → ( 𝑥 ∈ 𝑦 → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑦 ) ) ) )
28 27 ralrimivv ⊢ ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) → ∀ 𝑥 ∈ dom 𝐹 ∀ 𝑦 ∈ dom 𝐹 ( 𝑥 ∈ 𝑦 → ( 𝐹 ‘ 𝑥 ) ∈ ( 𝐹 ‘ 𝑦 ) ) )
29 28 3ad2ant1 ⊢ ( ( 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 ⊢ ( 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 ⊢ ( ( 𝐹 Isom E , E ( 𝐴 , 𝐵 ) ∧ Ord 𝐴 ∧ 𝐵 ⊆ On ) → Smo 𝐹 )