Theorem dffin1-5

Description: Compact quantifier-free version of the standard definition df-fin . (Contributed by Stefan O'Rear, 6-Jan-2015)

	Ref	Expression
Assertion	dffin1-5	⊢ Fin = ( ≈ “ ω )

Proof

Step	Hyp	Ref	Expression
1		ensymb	⊢ ( 𝑥 ≈ 𝑦 ↔ 𝑦 ≈ 𝑥 )
2	1	rexbii	⊢ ( ∃ 𝑦 ∈ ω 𝑥 ≈ 𝑦 ↔ ∃ 𝑦 ∈ ω 𝑦 ≈ 𝑥 )
3	2	abbii	⊢ { 𝑥 ∣ ∃ 𝑦 ∈ ω 𝑥 ≈ 𝑦 } = { 𝑥 ∣ ∃ 𝑦 ∈ ω 𝑦 ≈ 𝑥 }
4		df-fin	⊢ Fin = { 𝑥 ∣ ∃ 𝑦 ∈ ω 𝑥 ≈ 𝑦 }
5		dfima2	⊢ ( ≈ “ ω ) = { 𝑥 ∣ ∃ 𝑦 ∈ ω 𝑦 ≈ 𝑥 }
6	3 4 5	3eqtr4i	⊢ Fin = ( ≈ “ ω )