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Metamath Proof Explorer

Theorem f1oen4g

Description: The domain and range of a one-to-one, onto set function are equinumerous. This variation of f1oeng does not require the Axiom of Replacement nor the Axiom of Power Sets nor the Axiom of Union. (Contributed by BTernaryTau, 7-Dec-2024)

Ref Expression
Assertion f1oen4g ( ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 )

Proof

Step Hyp Ref Expression
1 f1oeq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) )
2 1 spcegv ⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) )
3 2 imp ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 )
4 3 3ad2antl1 ⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 )
5 breng ⊢ ( ( 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ≈ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) )
6 5 3adant1 ⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) → ( 𝐴 ≈ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) )
7 6 adantr ⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → ( 𝐴 ≈ 𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴 –1-1-onto→ 𝐵 ) )
8 4 7 mpbird ⊢ ( ( ( 𝐹 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋 ) ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → 𝐴 ≈ 𝐵 )