Metamath Proof Explorer


Theorem isf34lem3

Description: Lemma for isfin3-4 . (Contributed by Stefan O'Rear, 7-Nov-2014) (Revised by Mario Carneiro, 17-May-2015)

Ref Expression
Hypothesis compss.a 𝐹 = ( 𝑥 ∈ 𝒫 𝐴 ↦ ( 𝐴𝑥 ) )
Assertion isf34lem3 ( ( 𝐴𝑉𝑋 ⊆ 𝒫 𝐴 ) → ( 𝐹 “ ( 𝐹𝑋 ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 compss.a 𝐹 = ( 𝑥 ∈ 𝒫 𝐴 ↦ ( 𝐴𝑥 ) )
2 1 compsscnv 𝐹 = 𝐹
3 2 imaeq1i ( 𝐹 “ ( 𝐹𝑋 ) ) = ( 𝐹 “ ( 𝐹𝑋 ) )
4 1 compssiso ( 𝐴𝑉𝐹 Isom [] , [] ( 𝒫 𝐴 , 𝒫 𝐴 ) )
5 isof1o ( 𝐹 Isom [] , [] ( 𝒫 𝐴 , 𝒫 𝐴 ) → 𝐹 : 𝒫 𝐴1-1-onto→ 𝒫 𝐴 )
6 f1of1 ( 𝐹 : 𝒫 𝐴1-1-onto→ 𝒫 𝐴𝐹 : 𝒫 𝐴1-1→ 𝒫 𝐴 )
7 4 5 6 3syl ( 𝐴𝑉𝐹 : 𝒫 𝐴1-1→ 𝒫 𝐴 )
8 f1imacnv ( ( 𝐹 : 𝒫 𝐴1-1→ 𝒫 𝐴𝑋 ⊆ 𝒫 𝐴 ) → ( 𝐹 “ ( 𝐹𝑋 ) ) = 𝑋 )
9 7 8 sylan ( ( 𝐴𝑉𝑋 ⊆ 𝒫 𝐴 ) → ( 𝐹 “ ( 𝐹𝑋 ) ) = 𝑋 )
10 3 9 eqtr3id ( ( 𝐴𝑉𝑋 ⊆ 𝒫 𝐴 ) → ( 𝐹 “ ( 𝐹𝑋 ) ) = 𝑋 )