Theorem smoiun

Description: The value of a strictly monotone ordinal function contains its indexed union. (Contributed by Andrew Salmon, 22-Nov-2011)

	Ref	Expression
Assertion	smoiun	⊢ ( ( Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ) → ∪ 𝑥 ∈ 𝐴 ( 𝐵 ‘ 𝑥 ) ⊆ ( 𝐵 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ‘ 𝑥 ) ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ‘ 𝑥 ) )
2		smofvon	⊢ ( ( Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ) → ( 𝐵 ‘ 𝐴 ) ∈ On )
3		smoel	⊢ ( ( Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 ‘ 𝑥 ) ∈ ( 𝐵 ‘ 𝐴 ) )
4	3	3expia	⊢ ( ( Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝐵 ‘ 𝑥 ) ∈ ( 𝐵 ‘ 𝐴 ) ) )
5		ontr1	⊢ ( ( 𝐵 ‘ 𝐴 ) ∈ On → ( ( 𝑦 ∈ ( 𝐵 ‘ 𝑥 ) ∧ ( 𝐵 ‘ 𝑥 ) ∈ ( 𝐵 ‘ 𝐴 ) ) → 𝑦 ∈ ( 𝐵 ‘ 𝐴 ) ) )
6	5	expcomd	⊢ ( ( 𝐵 ‘ 𝐴 ) ∈ On → ( ( 𝐵 ‘ 𝑥 ) ∈ ( 𝐵 ‘ 𝐴 ) → ( 𝑦 ∈ ( 𝐵 ‘ 𝑥 ) → 𝑦 ∈ ( 𝐵 ‘ 𝐴 ) ) ) )
7	2 4 6	sylsyld	⊢ ( ( Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ) → ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ ( 𝐵 ‘ 𝑥 ) → 𝑦 ∈ ( 𝐵 ‘ 𝐴 ) ) ) )
8	7	rexlimdv	⊢ ( ( Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ) → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ ( 𝐵 ‘ 𝑥 ) → 𝑦 ∈ ( 𝐵 ‘ 𝐴 ) ) )
9	1 8	biimtrid	⊢ ( ( Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ) → ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ( 𝐵 ‘ 𝑥 ) → 𝑦 ∈ ( 𝐵 ‘ 𝐴 ) ) )
10	9	ssrdv	⊢ ( ( Smo 𝐵 ∧ 𝐴 ∈ dom 𝐵 ) → ∪ 𝑥 ∈ 𝐴 ( 𝐵 ‘ 𝑥 ) ⊆ ( 𝐵 ‘ 𝐴 ) )