Step |
Hyp |
Ref |
Expression |
1 |
|
opsrbas.s |
⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 ) |
2 |
|
opsrbas.o |
⊢ 𝑂 = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 ) |
3 |
|
opsrbas.t |
⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐼 × 𝐼 ) ) |
4 |
|
opsrbaslem.1 |
⊢ 𝐸 = Slot 𝑁 |
5 |
|
opsrbaslem.2 |
⊢ 𝑁 ∈ ℕ |
6 |
|
opsrbaslem.3 |
⊢ 𝑁 < ; 1 0 |
7 |
4 5
|
ndxid |
⊢ 𝐸 = Slot ( 𝐸 ‘ ndx ) |
8 |
5
|
nnrei |
⊢ 𝑁 ∈ ℝ |
9 |
8 6
|
ltneii |
⊢ 𝑁 ≠ ; 1 0 |
10 |
4 5
|
ndxarg |
⊢ ( 𝐸 ‘ ndx ) = 𝑁 |
11 |
|
plendx |
⊢ ( le ‘ ndx ) = ; 1 0 |
12 |
10 11
|
neeq12i |
⊢ ( ( 𝐸 ‘ ndx ) ≠ ( le ‘ ndx ) ↔ 𝑁 ≠ ; 1 0 ) |
13 |
9 12
|
mpbir |
⊢ ( 𝐸 ‘ ndx ) ≠ ( le ‘ ndx ) |
14 |
7 13
|
setsnid |
⊢ ( 𝐸 ‘ 𝑆 ) = ( 𝐸 ‘ ( 𝑆 sSet 〈 ( le ‘ ndx ) , ( le ‘ 𝑂 ) 〉 ) ) |
15 |
|
eqid |
⊢ ( le ‘ 𝑂 ) = ( le ‘ 𝑂 ) |
16 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝐼 ∈ V ) |
17 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝑅 ∈ V ) |
18 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝑇 ⊆ ( 𝐼 × 𝐼 ) ) |
19 |
1 2 15 16 17 18
|
opsrval2 |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝑂 = ( 𝑆 sSet 〈 ( le ‘ ndx ) , ( le ‘ 𝑂 ) 〉 ) ) |
20 |
19
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝐸 ‘ 𝑂 ) = ( 𝐸 ‘ ( 𝑆 sSet 〈 ( le ‘ ndx ) , ( le ‘ 𝑂 ) 〉 ) ) ) |
21 |
14 20
|
eqtr4id |
⊢ ( ( 𝜑 ∧ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝐸 ‘ 𝑆 ) = ( 𝐸 ‘ 𝑂 ) ) |
22 |
|
0fv |
⊢ ( ∅ ‘ 𝑇 ) = ∅ |
23 |
22
|
eqcomi |
⊢ ∅ = ( ∅ ‘ 𝑇 ) |
24 |
|
reldmpsr |
⊢ Rel dom mPwSer |
25 |
24
|
ovprc |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mPwSer 𝑅 ) = ∅ ) |
26 |
|
reldmopsr |
⊢ Rel dom ordPwSer |
27 |
26
|
ovprc |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 ordPwSer 𝑅 ) = ∅ ) |
28 |
27
|
fveq1d |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 ) = ( ∅ ‘ 𝑇 ) ) |
29 |
23 25 28
|
3eqtr4a |
⊢ ( ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) → ( 𝐼 mPwSer 𝑅 ) = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 ) ) |
30 |
29
|
adantl |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝐼 mPwSer 𝑅 ) = ( ( 𝐼 ordPwSer 𝑅 ) ‘ 𝑇 ) ) |
31 |
30 1 2
|
3eqtr4g |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → 𝑆 = 𝑂 ) |
32 |
31
|
fveq2d |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 ∈ V ∧ 𝑅 ∈ V ) ) → ( 𝐸 ‘ 𝑆 ) = ( 𝐸 ‘ 𝑂 ) ) |
33 |
21 32
|
pm2.61dan |
⊢ ( 𝜑 → ( 𝐸 ‘ 𝑆 ) = ( 𝐸 ‘ 𝑂 ) ) |