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Metamath Proof Explorer

Theorem enssdomOLD

Description: Obsolete version of enssdom as of 10-Feb-2026. (Contributed by NM, 31-Mar-1998) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion enssdomOLD ≈ ⊆ ≼

Proof

Step Hyp Ref Expression
1 relen ⊢ Rel ≈
2 f1of1 ⊢ ( 𝑓 : 𝑥 –1-1-onto→ 𝑦 → 𝑓 : 𝑥 –1-1→ 𝑦 )
3 2 eximi ⊢ ( ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑦 → ∃ 𝑓 𝑓 : 𝑥 –1-1→ 𝑦 )
4 opabidw ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑦 } ↔ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑦 )
5 opabidw ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1→ 𝑦 } ↔ ∃ 𝑓 𝑓 : 𝑥 –1-1→ 𝑦 )
6 3 4 5 3imtr4i ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑦 } → ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1→ 𝑦 } )
7 df-en ⊢ ≈ = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑦 }
8 7 eleq2i ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ≈ ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1-onto→ 𝑦 } )
9 df-dom ⊢ ≼ = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1→ 𝑦 }
10 9 eleq2i ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ≼ ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥 –1-1→ 𝑦 } )
11 6 8 10 3imtr4i ⊢ ( ⟨ 𝑥 , 𝑦 ⟩ ∈ ≈ → ⟨ 𝑥 , 𝑦 ⟩ ∈ ≼ )
12 1 11 relssi ⊢ ≈ ⊆ ≼