Theorem smorndom

Description: The range of a strictly monotone ordinal function dominates the domain. (Contributed by Mario Carneiro, 13-Mar-2013)

	Ref	Expression
Assertion	smorndom	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) → 𝐴 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		simpl1	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐹 : 𝐴 ⟶ 𝐵 )
2	1	ffnd	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐹 Fn 𝐴 )
3		simpl2	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → Smo 𝐹 )
4		smodm2	⊢ ( ( 𝐹 Fn 𝐴 ∧ Smo 𝐹 ) → Ord 𝐴 )
5	2 3 4	syl2anc	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → Ord 𝐴 )
6		ordelord	⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ 𝐴 ) → Ord 𝑥 )
7	5 6	sylancom	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → Ord 𝑥 )
8		simpl3	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → Ord 𝐵 )
9		simpr	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 )
10		smogt	⊢ ( ( 𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) )
11	2 3 9 10	syl3anc	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) )
12		ffvelrn	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
13	12	3ad2antl1	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
14		ordtr2	⊢ ( ( Ord 𝑥 ∧ Ord 𝐵 ) → ( ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) )
15	14	imp	⊢ ( ( ( Ord 𝑥 ∧ Ord 𝐵 ) ∧ ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 )
16	7 8 11 13 15	syl22anc	⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐵 )
17	16	ex	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) )
18	17	ssrdv	⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) → 𝐴 ⊆ 𝐵 )