Metamath Proof Explorer


Theorem wfrfun

Description: The well-ordered function generator generates a function. (Contributed by Scott Fenton, 21-Apr-2011) (Revised by Mario Carneiro, 26-Jun-2015)

Ref Expression
Hypotheses wfrfun.1 𝑅 We 𝐴
wfrfun.2 𝑅 Se 𝐴
wfrfun.3 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 )
Assertion wfrfun Fun 𝐹

Proof

Step Hyp Ref Expression
1 wfrfun.1 𝑅 We 𝐴
2 wfrfun.2 𝑅 Se 𝐴
3 wfrfun.3 𝐹 = wrecs ( 𝑅 , 𝐴 , 𝐺 )
4 3 wfrrel Rel 𝐹
5 df-wrecs wrecs ( 𝑅 , 𝐴 , 𝐺 ) = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
6 3 5 eqtri 𝐹 = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
7 6 eleq2i ( ⟨ 𝑥 , 𝑢 ⟩ ∈ 𝐹 ↔ ⟨ 𝑥 , 𝑢 ⟩ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } )
8 eluni ( ⟨ 𝑥 , 𝑢 ⟩ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ↔ ∃ 𝑔 ( ⟨ 𝑥 , 𝑢 ⟩ ∈ 𝑔𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) )
9 7 8 bitri ( ⟨ 𝑥 , 𝑢 ⟩ ∈ 𝐹 ↔ ∃ 𝑔 ( ⟨ 𝑥 , 𝑢 ⟩ ∈ 𝑔𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) )
10 df-br ( 𝑥 𝐹 𝑢 ↔ ⟨ 𝑥 , 𝑢 ⟩ ∈ 𝐹 )
11 df-br ( 𝑥 𝑔 𝑢 ↔ ⟨ 𝑥 , 𝑢 ⟩ ∈ 𝑔 )
12 11 anbi1i ( ( 𝑥 𝑔 𝑢𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ↔ ( ⟨ 𝑥 , 𝑢 ⟩ ∈ 𝑔𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) )
13 12 exbii ( ∃ 𝑔 ( 𝑥 𝑔 𝑢𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ↔ ∃ 𝑔 ( ⟨ 𝑥 , 𝑢 ⟩ ∈ 𝑔𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) )
14 9 10 13 3bitr4i ( 𝑥 𝐹 𝑢 ↔ ∃ 𝑔 ( 𝑥 𝑔 𝑢𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) )
15 6 eleq2i ( ⟨ 𝑥 , 𝑣 ⟩ ∈ 𝐹 ↔ ⟨ 𝑥 , 𝑣 ⟩ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } )
16 eluni ( ⟨ 𝑥 , 𝑣 ⟩ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ↔ ∃ ( ⟨ 𝑥 , 𝑣 ⟩ ∈ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) )
17 15 16 bitri ( ⟨ 𝑥 , 𝑣 ⟩ ∈ 𝐹 ↔ ∃ ( ⟨ 𝑥 , 𝑣 ⟩ ∈ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) )
18 df-br ( 𝑥 𝐹 𝑣 ↔ ⟨ 𝑥 , 𝑣 ⟩ ∈ 𝐹 )
19 df-br ( 𝑥 𝑣 ↔ ⟨ 𝑥 , 𝑣 ⟩ ∈ )
20 19 anbi1i ( ( 𝑥 𝑣 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ↔ ( ⟨ 𝑥 , 𝑣 ⟩ ∈ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) )
21 20 exbii ( ∃ ( 𝑥 𝑣 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ↔ ∃ ( ⟨ 𝑥 , 𝑣 ⟩ ∈ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) )
22 17 18 21 3bitr4i ( 𝑥 𝐹 𝑣 ↔ ∃ ( 𝑥 𝑣 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) )
23 14 22 anbi12i ( ( 𝑥 𝐹 𝑢𝑥 𝐹 𝑣 ) ↔ ( ∃ 𝑔 ( 𝑥 𝑔 𝑢𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ∧ ∃ ( 𝑥 𝑣 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ) )
24 exdistrv ( ∃ 𝑔 ( ( 𝑥 𝑔 𝑢𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ∧ ( 𝑥 𝑣 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ) ↔ ( ∃ 𝑔 ( 𝑥 𝑔 𝑢𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ∧ ∃ ( 𝑥 𝑣 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ) )
25 23 24 bitr4i ( ( 𝑥 𝐹 𝑢𝑥 𝐹 𝑣 ) ↔ ∃ 𝑔 ( ( 𝑥 𝑔 𝑢𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ∧ ( 𝑥 𝑣 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ) )
26 eqid { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } = { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) }
27 1 2 26 wfrlem5 ( ( 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ∧ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) → ( ( 𝑥 𝑔 𝑢𝑥 𝑣 ) → 𝑢 = 𝑣 ) )
28 27 impcom ( ( ( 𝑥 𝑔 𝑢𝑥 𝑣 ) ∧ ( 𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ∧ ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ) → 𝑢 = 𝑣 )
29 28 an4s ( ( ( 𝑥 𝑔 𝑢𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ∧ ( 𝑥 𝑣 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ) → 𝑢 = 𝑣 )
30 29 exlimivv ( ∃ 𝑔 ( ( 𝑥 𝑔 𝑢𝑔 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ∧ ( 𝑥 𝑣 ∈ { 𝑓 ∣ ∃ 𝑥 ( 𝑓 Fn 𝑥 ∧ ( 𝑥𝐴 ∧ ∀ 𝑦𝑥 Pred ( 𝑅 , 𝐴 , 𝑦 ) ⊆ 𝑥 ) ∧ ∀ 𝑦𝑥 ( 𝑓𝑦 ) = ( 𝐺 ‘ ( 𝑓 ↾ Pred ( 𝑅 , 𝐴 , 𝑦 ) ) ) ) } ) ) → 𝑢 = 𝑣 )
31 25 30 sylbi ( ( 𝑥 𝐹 𝑢𝑥 𝐹 𝑣 ) → 𝑢 = 𝑣 )
32 31 ax-gen 𝑣 ( ( 𝑥 𝐹 𝑢𝑥 𝐹 𝑣 ) → 𝑢 = 𝑣 )
33 32 gen2 𝑥𝑢𝑣 ( ( 𝑥 𝐹 𝑢𝑥 𝐹 𝑣 ) → 𝑢 = 𝑣 )
34 dffun2 ( Fun 𝐹 ↔ ( Rel 𝐹 ∧ ∀ 𝑥𝑢𝑣 ( ( 𝑥 𝐹 𝑢𝑥 𝐹 𝑣 ) → 𝑢 = 𝑣 ) ) )
35 4 33 34 mpbir2an Fun 𝐹