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 ) |
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 ) |