Database
ZF (ZERMELO-FRAENKEL) SET THEORY
ZF Set Theory - add the Axiom of Infinity
Transitive closure under a relationship
trpredeq1
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trpredeq2
Metamath Proof Explorer
Ascii
Unicode
Theorem
trpredeq1
Description:
Equality theorem for transitive predecessors.
(Contributed by
Scott Fenton
, 2-Feb-2011)
Ref
Expression
Assertion
trpredeq1
⊢
R
=
S
→
TrPred
R
A
X
=
TrPred
S
A
X
Proof
Step
Hyp
Ref
Expression
1
predeq1
⊢
R
=
S
→
Pred
R
A
y
=
Pred
S
A
y
2
1
iuneq2d
⊢
R
=
S
→
⋃
y
∈
a
Pred
R
A
y
=
⋃
y
∈
a
Pred
S
A
y
3
2
mpteq2dv
⊢
R
=
S
→
a
∈
V
⟼
⋃
y
∈
a
Pred
R
A
y
=
a
∈
V
⟼
⋃
y
∈
a
Pred
S
A
y
4
predeq1
⊢
R
=
S
→
Pred
R
A
X
=
Pred
S
A
X
5
rdgeq12
⊢
a
∈
V
⟼
⋃
y
∈
a
Pred
R
A
y
=
a
∈
V
⟼
⋃
y
∈
a
Pred
S
A
y
∧
Pred
R
A
X
=
Pred
S
A
X
→
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
R
A
y
Pred
R
A
X
=
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
S
A
y
Pred
S
A
X
6
3
4
5
syl2anc
⊢
R
=
S
→
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
R
A
y
Pred
R
A
X
=
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
S
A
y
Pred
S
A
X
7
6
reseq1d
⊢
R
=
S
→
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
R
A
y
Pred
R
A
X
↾
ω
=
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
S
A
y
Pred
S
A
X
↾
ω
8
7
rneqd
⊢
R
=
S
→
ran
⁡
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
R
A
y
Pred
R
A
X
↾
ω
=
ran
⁡
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
S
A
y
Pred
S
A
X
↾
ω
9
8
unieqd
⊢
R
=
S
→
⋃
ran
⁡
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
R
A
y
Pred
R
A
X
↾
ω
=
⋃
ran
⁡
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
S
A
y
Pred
S
A
X
↾
ω
10
df-trpred
⊢
TrPred
R
A
X
=
⋃
ran
⁡
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
R
A
y
Pred
R
A
X
↾
ω
11
df-trpred
⊢
TrPred
S
A
X
=
⋃
ran
⁡
rec
⁡
a
∈
V
⟼
⋃
y
∈
a
Pred
S
A
y
Pred
S
A
X
↾
ω
12
9
10
11
3eqtr4g
⊢
R
=
S
→
TrPred
R
A
X
=
TrPred
S
A
X