Metamath Proof Explorer


Theorem brenOLD

Description: Obsolete version of bren as of 23-Sep-2024. (Contributed by NM, 15-Jun-1998) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion brenOLD ( 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 f1oeq2 ( 𝑥 = 𝐴 → ( 𝑓 : 𝑥1-1-onto𝑦𝑓 : 𝐴1-1-onto𝑦 ) )
16 15 exbidv ( 𝑥 = 𝐴 → ( ∃ 𝑓 𝑓 : 𝑥1-1-onto𝑦 ↔ ∃ 𝑓 𝑓 : 𝐴1-1-onto𝑦 ) )
17 f1oeq3 ( 𝑦 = 𝐵 → ( 𝑓 : 𝐴1-1-onto𝑦𝑓 : 𝐴1-1-onto𝐵 ) )
18 17 exbidv ( 𝑦 = 𝐵 → ( ∃ 𝑓 𝑓 : 𝐴1-1-onto𝑦 ↔ ∃ 𝑓 𝑓 : 𝐴1-1-onto𝐵 ) )
19 df-en ≈ = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑓 𝑓 : 𝑥1-1-onto𝑦 }
20 16 18 19 brabg ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴1-1-onto𝐵 ) )
21 1 14 20 pm5.21nii ( 𝐴𝐵 ↔ ∃ 𝑓 𝑓 : 𝐴1-1-onto𝐵 )