Database
SUPPLEMENTARY MATERIAL (USERS' MATHBOXES)
Mathbox for Peter Mazsa
Functions
elfunsALTV3
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elfunsALTV4
Metamath Proof Explorer
Ascii
Unicode
Theorem
elfunsALTV3
Description:
Elementhood in the class of functions.
(Contributed by
Peter Mazsa
, 31-Aug-2021)
Ref
Expression
Assertion
elfunsALTV3
⊢
F
∈
FunsALTV
↔
∀
u
∀
x
∀
y
u
F
x
∧
u
F
y
→
x
=
y
∧
F
∈
Rels
Proof
Step
Hyp
Ref
Expression
1
elfunsALTV
⊢
F
∈
FunsALTV
↔
≀
F
∈
CnvRefRels
∧
F
∈
Rels
2
cosselcnvrefrels3
⊢
≀
F
∈
CnvRefRels
↔
∀
u
∀
x
∀
y
u
F
x
∧
u
F
y
→
x
=
y
∧
≀
F
∈
Rels
3
cosselrels
⊢
F
∈
Rels
→
≀
F
∈
Rels
4
3
biantrud
⊢
F
∈
Rels
→
∀
u
∀
x
∀
y
u
F
x
∧
u
F
y
→
x
=
y
↔
∀
u
∀
x
∀
y
u
F
x
∧
u
F
y
→
x
=
y
∧
≀
F
∈
Rels
5
2
4
bitr4id
⊢
F
∈
Rels
→
≀
F
∈
CnvRefRels
↔
∀
u
∀
x
∀
y
u
F
x
∧
u
F
y
→
x
=
y
6
5
pm5.32ri
⊢
≀
F
∈
CnvRefRels
∧
F
∈
Rels
↔
∀
u
∀
x
∀
y
u
F
x
∧
u
F
y
→
x
=
y
∧
F
∈
Rels
7
1
6
bitri
⊢
F
∈
FunsALTV
↔
∀
u
∀
x
∀
y
u
F
x
∧
u
F
y
→
x
=
y
∧
F
∈
Rels