Step |
Hyp |
Ref |
Expression |
1 |
|
om1val.o |
⊢ 𝑂 = ( 𝐽 Ω1 𝑌 ) |
2 |
|
om1val.b |
⊢ ( 𝜑 → 𝐵 = { 𝑓 ∈ ( II Cn 𝐽 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑌 ∧ ( 𝑓 ‘ 1 ) = 𝑌 ) } ) |
3 |
|
om1val.p |
⊢ ( 𝜑 → + = ( *𝑝 ‘ 𝐽 ) ) |
4 |
|
om1val.k |
⊢ ( 𝜑 → 𝐾 = ( 𝐽 ↑ko II ) ) |
5 |
|
om1val.j |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
6 |
|
om1val.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑋 ) |
7 |
|
df-om1 |
⊢ Ω1 = ( 𝑗 ∈ Top , 𝑦 ∈ ∪ 𝑗 ↦ { 〈 ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } 〉 , 〈 ( +g ‘ ndx ) , ( *𝑝 ‘ 𝑗 ) 〉 , 〈 ( TopSet ‘ ndx ) , ( 𝑗 ↑ko II ) 〉 } ) |
8 |
7
|
a1i |
⊢ ( 𝜑 → Ω1 = ( 𝑗 ∈ Top , 𝑦 ∈ ∪ 𝑗 ↦ { 〈 ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } 〉 , 〈 ( +g ‘ ndx ) , ( *𝑝 ‘ 𝑗 ) 〉 , 〈 ( TopSet ‘ ndx ) , ( 𝑗 ↑ko II ) 〉 } ) ) |
9 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → 𝑗 = 𝐽 ) |
10 |
9
|
oveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( II Cn 𝑗 ) = ( II Cn 𝐽 ) ) |
11 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → 𝑦 = 𝑌 ) |
12 |
11
|
eqeq2d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑓 ‘ 0 ) = 𝑦 ↔ ( 𝑓 ‘ 0 ) = 𝑌 ) ) |
13 |
11
|
eqeq2d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( ( 𝑓 ‘ 1 ) = 𝑦 ↔ ( 𝑓 ‘ 1 ) = 𝑌 ) ) |
14 |
12 13
|
anbi12d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) ↔ ( ( 𝑓 ‘ 0 ) = 𝑌 ∧ ( 𝑓 ‘ 1 ) = 𝑌 ) ) ) |
15 |
10 14
|
rabeqbidv |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } = { 𝑓 ∈ ( II Cn 𝐽 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑌 ∧ ( 𝑓 ‘ 1 ) = 𝑌 ) } ) |
16 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → 𝐵 = { 𝑓 ∈ ( II Cn 𝐽 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑌 ∧ ( 𝑓 ‘ 1 ) = 𝑌 ) } ) |
17 |
15 16
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } = 𝐵 ) |
18 |
17
|
opeq2d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → 〈 ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } 〉 = 〈 ( Base ‘ ndx ) , 𝐵 〉 ) |
19 |
9
|
fveq2d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( *𝑝 ‘ 𝑗 ) = ( *𝑝 ‘ 𝐽 ) ) |
20 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → + = ( *𝑝 ‘ 𝐽 ) ) |
21 |
19 20
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( *𝑝 ‘ 𝑗 ) = + ) |
22 |
21
|
opeq2d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → 〈 ( +g ‘ ndx ) , ( *𝑝 ‘ 𝑗 ) 〉 = 〈 ( +g ‘ ndx ) , + 〉 ) |
23 |
9
|
oveq1d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( 𝑗 ↑ko II ) = ( 𝐽 ↑ko II ) ) |
24 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → 𝐾 = ( 𝐽 ↑ko II ) ) |
25 |
23 24
|
eqtr4d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → ( 𝑗 ↑ko II ) = 𝐾 ) |
26 |
25
|
opeq2d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → 〈 ( TopSet ‘ ndx ) , ( 𝑗 ↑ko II ) 〉 = 〈 ( TopSet ‘ ndx ) , 𝐾 〉 ) |
27 |
18 22 26
|
tpeq123d |
⊢ ( ( 𝜑 ∧ ( 𝑗 = 𝐽 ∧ 𝑦 = 𝑌 ) ) → { 〈 ( Base ‘ ndx ) , { 𝑓 ∈ ( II Cn 𝑗 ) ∣ ( ( 𝑓 ‘ 0 ) = 𝑦 ∧ ( 𝑓 ‘ 1 ) = 𝑦 ) } 〉 , 〈 ( +g ‘ ndx ) , ( *𝑝 ‘ 𝑗 ) 〉 , 〈 ( TopSet ‘ ndx ) , ( 𝑗 ↑ko II ) 〉 } = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , + 〉 , 〈 ( TopSet ‘ ndx ) , 𝐾 〉 } ) |
28 |
|
unieq |
⊢ ( 𝑗 = 𝐽 → ∪ 𝑗 = ∪ 𝐽 ) |
29 |
28
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝐽 ) → ∪ 𝑗 = ∪ 𝐽 ) |
30 |
|
toponuni |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝑋 = ∪ 𝐽 ) |
31 |
5 30
|
syl |
⊢ ( 𝜑 → 𝑋 = ∪ 𝐽 ) |
32 |
31
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝐽 ) → 𝑋 = ∪ 𝐽 ) |
33 |
29 32
|
eqtr4d |
⊢ ( ( 𝜑 ∧ 𝑗 = 𝐽 ) → ∪ 𝑗 = 𝑋 ) |
34 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
35 |
5 34
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
36 |
|
tpex |
⊢ { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , + 〉 , 〈 ( TopSet ‘ ndx ) , 𝐾 〉 } ∈ V |
37 |
36
|
a1i |
⊢ ( 𝜑 → { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , + 〉 , 〈 ( TopSet ‘ ndx ) , 𝐾 〉 } ∈ V ) |
38 |
8 27 33 35 6 37
|
ovmpodx |
⊢ ( 𝜑 → ( 𝐽 Ω1 𝑌 ) = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , + 〉 , 〈 ( TopSet ‘ ndx ) , 𝐾 〉 } ) |
39 |
1 38
|
syl5eq |
⊢ ( 𝜑 → 𝑂 = { 〈 ( Base ‘ ndx ) , 𝐵 〉 , 〈 ( +g ‘ ndx ) , + 〉 , 〈 ( TopSet ‘ ndx ) , 𝐾 〉 } ) |