Metamath Proof Explorer


Theorem marypha2lem3

Description: Lemma for marypha2 . Properties of the used relation. (Contributed by Stefan O'Rear, 20-Feb-2015)

Ref Expression
Hypothesis marypha2lem.t 𝑇 = 𝑥𝐴 ( { 𝑥 } × ( 𝐹𝑥 ) )
Assertion marypha2lem3 ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐺𝑇 ↔ ∀ 𝑥𝐴 ( 𝐺𝑥 ) ∈ ( 𝐹𝑥 ) ) )

Proof

Step Hyp Ref Expression
1 marypha2lem.t 𝑇 = 𝑥𝐴 ( { 𝑥 } × ( 𝐹𝑥 ) )
2 dffn5 ( 𝐺 Fn 𝐴𝐺 = ( 𝑥𝐴 ↦ ( 𝐺𝑥 ) ) )
3 2 biimpi ( 𝐺 Fn 𝐴𝐺 = ( 𝑥𝐴 ↦ ( 𝐺𝑥 ) ) )
4 3 adantl ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → 𝐺 = ( 𝑥𝐴 ↦ ( 𝐺𝑥 ) ) )
5 df-mpt ( 𝑥𝐴 ↦ ( 𝐺𝑥 ) ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) }
6 4 5 eqtrdi ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → 𝐺 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) } )
7 1 marypha2lem2 𝑇 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) }
8 7 a1i ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → 𝑇 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) } )
9 6 8 sseq12d ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐺𝑇 ↔ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) } ) )
10 ssopab2bw ( { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) } ⊆ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) } ↔ ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) → ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) ) )
11 9 10 bitrdi ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐺𝑇 ↔ ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) → ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) ) ) )
12 19.21v ( ∀ 𝑦 ( 𝑥𝐴 → ( 𝑦 = ( 𝐺𝑥 ) → 𝑦 ∈ ( 𝐹𝑥 ) ) ) ↔ ( 𝑥𝐴 → ∀ 𝑦 ( 𝑦 = ( 𝐺𝑥 ) → 𝑦 ∈ ( 𝐹𝑥 ) ) ) )
13 imdistan ( ( 𝑥𝐴 → ( 𝑦 = ( 𝐺𝑥 ) → 𝑦 ∈ ( 𝐹𝑥 ) ) ) ↔ ( ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) → ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) ) )
14 13 albii ( ∀ 𝑦 ( 𝑥𝐴 → ( 𝑦 = ( 𝐺𝑥 ) → 𝑦 ∈ ( 𝐹𝑥 ) ) ) ↔ ∀ 𝑦 ( ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) → ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) ) )
15 fvex ( 𝐺𝑥 ) ∈ V
16 eleq1 ( 𝑦 = ( 𝐺𝑥 ) → ( 𝑦 ∈ ( 𝐹𝑥 ) ↔ ( 𝐺𝑥 ) ∈ ( 𝐹𝑥 ) ) )
17 15 16 ceqsalv ( ∀ 𝑦 ( 𝑦 = ( 𝐺𝑥 ) → 𝑦 ∈ ( 𝐹𝑥 ) ) ↔ ( 𝐺𝑥 ) ∈ ( 𝐹𝑥 ) )
18 17 imbi2i ( ( 𝑥𝐴 → ∀ 𝑦 ( 𝑦 = ( 𝐺𝑥 ) → 𝑦 ∈ ( 𝐹𝑥 ) ) ) ↔ ( 𝑥𝐴 → ( 𝐺𝑥 ) ∈ ( 𝐹𝑥 ) ) )
19 12 14 18 3bitr3i ( ∀ 𝑦 ( ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) → ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) ) ↔ ( 𝑥𝐴 → ( 𝐺𝑥 ) ∈ ( 𝐹𝑥 ) ) )
20 19 albii ( ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) → ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) ) ↔ ∀ 𝑥 ( 𝑥𝐴 → ( 𝐺𝑥 ) ∈ ( 𝐹𝑥 ) ) )
21 df-ral ( ∀ 𝑥𝐴 ( 𝐺𝑥 ) ∈ ( 𝐹𝑥 ) ↔ ∀ 𝑥 ( 𝑥𝐴 → ( 𝐺𝑥 ) ∈ ( 𝐹𝑥 ) ) )
22 20 21 bitr4i ( ∀ 𝑥𝑦 ( ( 𝑥𝐴𝑦 = ( 𝐺𝑥 ) ) → ( 𝑥𝐴𝑦 ∈ ( 𝐹𝑥 ) ) ) ↔ ∀ 𝑥𝐴 ( 𝐺𝑥 ) ∈ ( 𝐹𝑥 ) )
23 11 22 bitrdi ( ( 𝐹 Fn 𝐴𝐺 Fn 𝐴 ) → ( 𝐺𝑇 ↔ ∀ 𝑥𝐴 ( 𝐺𝑥 ) ∈ ( 𝐹𝑥 ) ) )