Metamath Proof Explorer


Definition df-funs

Description: Define the class of all functions. See elfuns for membership. (Contributed by Scott Fenton, 18-Feb-2013)

Ref Expression
Assertion df-funs Funs = ( 𝒫 ( V × V ) ∖ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ E ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cfuns Funs
1 cvv V
2 1 1 cxp ( V × V )
3 2 cpw 𝒫 ( V × V )
4 cep E
5 c1st 1st
6 cid I
7 1 6 cdif ( V ∖ I )
8 c2nd 2nd
9 7 8 ccom ( ( V ∖ I ) ∘ 2nd )
10 5 9 ctxp ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) )
11 4 ccnv E
12 10 11 ccom ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ E )
13 4 12 ccom ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ E ) )
14 13 cfix Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ E ) )
15 3 14 cdif ( 𝒫 ( V × V ) ∖ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ E ) ) )
16 0 15 wceq Funs = ( 𝒫 ( V × V ) ∖ Fix ( E ∘ ( ( 1st ⊗ ( ( V ∖ I ) ∘ 2nd ) ) ∘ E ) ) )