Metamath Proof Explorer


Theorem bren

Description: Equinumerosity relation. (Contributed by NM, 15-Jun-1998) (Proof shortened by BTernaryTau, 23-Sep-2024)

Ref Expression
Assertion bren ( 𝐴𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴1-1-onto𝐵 )

Proof

Step Hyp Ref Expression
1 encv ( 𝐴𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
2 f1ofn ( 𝑓 : 𝐴1-1-onto𝐵𝑓 Fn 𝐴 )
3 fndm ( 𝑓 Fn 𝐴 → dom 𝑓 = 𝐴 )
4 vex 𝑓 ∈ V
5 4 dmex dom 𝑓 ∈ V
6 3 5 eqeltrrdi ( 𝑓 Fn 𝐴𝐴 ∈ V )
7 2 6 syl ( 𝑓 : 𝐴1-1-onto𝐵𝐴 ∈ V )
8 f1ofo ( 𝑓 : 𝐴1-1-onto𝐵𝑓 : 𝐴onto𝐵 )
9 forn ( 𝑓 : 𝐴onto𝐵 → ran 𝑓 = 𝐵 )
10 8 9 syl ( 𝑓 : 𝐴1-1-onto𝐵 → ran 𝑓 = 𝐵 )
11 4 rnex ran 𝑓 ∈ V
12 10 11 eqeltrrdi ( 𝑓 : 𝐴1-1-onto𝐵𝐵 ∈ V )
13 7 12 jca ( 𝑓 : 𝐴1-1-onto𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
14 13 exlimiv ( ∃ 𝑓 𝑓 : 𝐴1-1-onto𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
15 breng ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴1-1-onto𝐵 ) )
16 1 14 15 pm5.21nii ( 𝐴𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴1-1-onto𝐵 )