Theorem dfon2lem2

Description: Lemma for dfon2 . (Contributed by Scott Fenton, 28-Feb-2011)

	Ref	Expression
Assertion	dfon2lem2	⊢ ∪ { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓 ) } ⊆ 𝐴

Proof

Step	Hyp	Ref	Expression
1		simp1	⊢ ( ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓 ) → 𝑥 ⊆ 𝐴 )
2	1	ss2abi	⊢ { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓 ) } ⊆ { 𝑥 ∣ 𝑥 ⊆ 𝐴 }
3		df-pw	⊢ 𝒫 𝐴 = { 𝑥 ∣ 𝑥 ⊆ 𝐴 }
4	2 3	sseqtrri	⊢ { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓 ) } ⊆ 𝒫 𝐴
5		sspwuni	⊢ ( { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓 ) } ⊆ 𝒫 𝐴 ↔ ∪ { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓 ) } ⊆ 𝐴 )
6	4 5	mpbi	⊢ ∪ { 𝑥 ∣ ( 𝑥 ⊆ 𝐴 ∧ 𝜑 ∧ 𝜓 ) } ⊆ 𝐴