Step |
Hyp |
Ref |
Expression |
1 |
|
mptmpoopabbrd.g |
⊢ ( 𝜑 → 𝐺 ∈ 𝑊 ) |
2 |
|
mptmpoopabbrd.x |
⊢ ( 𝜑 → 𝑋 ∈ ( 𝐴 ‘ 𝐺 ) ) |
3 |
|
mptmpoopabbrd.y |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝐵 ‘ 𝐺 ) ) |
4 |
|
mptmpoopabbrd.1 |
⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝑌 ) → ( 𝜏 ↔ 𝜃 ) ) |
5 |
|
mptmpoopabbrd.2 |
⊢ ( 𝑔 = 𝐺 → ( 𝜒 ↔ 𝜏 ) ) |
6 |
|
mptmpoopabbrd.m |
⊢ 𝑀 = ( 𝑔 ∈ V ↦ ( 𝑎 ∈ ( 𝐴 ‘ 𝑔 ) , 𝑏 ∈ ( 𝐵 ‘ 𝑔 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜒 ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) } ) ) |
7 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( 𝐴 ‘ 𝑔 ) = ( 𝐴 ‘ 𝐺 ) ) |
8 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( 𝐵 ‘ 𝑔 ) = ( 𝐵 ‘ 𝐺 ) ) |
9 |
|
fveq2 |
⊢ ( 𝑔 = 𝐺 → ( 𝐷 ‘ 𝑔 ) = ( 𝐷 ‘ 𝐺 ) ) |
10 |
9
|
breqd |
⊢ ( 𝑔 = 𝐺 → ( 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ↔ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) ) |
11 |
5 10
|
anbi12d |
⊢ ( 𝑔 = 𝐺 → ( ( 𝜒 ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) ↔ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) ) ) |
12 |
11
|
opabbidv |
⊢ ( 𝑔 = 𝐺 → { 〈 𝑓 , ℎ 〉 ∣ ( 𝜒 ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) } = { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) |
13 |
7 8 12
|
mpoeq123dv |
⊢ ( 𝑔 = 𝐺 → ( 𝑎 ∈ ( 𝐴 ‘ 𝑔 ) , 𝑏 ∈ ( 𝐵 ‘ 𝑔 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜒 ∧ 𝑓 ( 𝐷 ‘ 𝑔 ) ℎ ) } ) = ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) ) |
14 |
1
|
elexd |
⊢ ( 𝜑 → 𝐺 ∈ V ) |
15 |
|
fvex |
⊢ ( 𝐴 ‘ 𝐺 ) ∈ V |
16 |
|
fvex |
⊢ ( 𝐵 ‘ 𝐺 ) ∈ V |
17 |
|
fvex |
⊢ ( 𝐷 ‘ 𝐺 ) ∈ V |
18 |
17
|
pwex |
⊢ 𝒫 ( 𝐷 ‘ 𝐺 ) ∈ V |
19 |
|
simpr |
⊢ ( ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) → 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) |
20 |
19
|
ssopab2i |
⊢ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ⊆ { 〈 𝑓 , ℎ 〉 ∣ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ } |
21 |
|
opabss |
⊢ { 〈 𝑓 , ℎ 〉 ∣ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ } ⊆ ( 𝐷 ‘ 𝐺 ) |
22 |
20 21
|
sstri |
⊢ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ⊆ ( 𝐷 ‘ 𝐺 ) |
23 |
17 22
|
elpwi2 |
⊢ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ∈ 𝒫 ( 𝐷 ‘ 𝐺 ) |
24 |
23
|
rgen2w |
⊢ ∀ 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) ∀ 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ∈ 𝒫 ( 𝐷 ‘ 𝐺 ) |
25 |
15 16 18 24
|
mpoexw |
⊢ ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) ∈ V |
26 |
25
|
a1i |
⊢ ( 𝜑 → ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) ∈ V ) |
27 |
6 13 14 26
|
fvmptd3 |
⊢ ( 𝜑 → ( 𝑀 ‘ 𝐺 ) = ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) ) |
28 |
27
|
oveqd |
⊢ ( 𝜑 → ( 𝑋 ( 𝑀 ‘ 𝐺 ) 𝑌 ) = ( 𝑋 ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) 𝑌 ) ) |
29 |
4
|
anbi1d |
⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝑌 ) → ( ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) ↔ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) ) ) |
30 |
29
|
opabbidv |
⊢ ( ( 𝑎 = 𝑋 ∧ 𝑏 = 𝑌 ) → { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } = { 〈 𝑓 , ℎ 〉 ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) |
31 |
|
eqid |
⊢ ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) = ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) |
32 |
|
ancom |
⊢ ( ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) ↔ ( 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ∧ 𝜃 ) ) |
33 |
32
|
opabbii |
⊢ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } = { 〈 𝑓 , ℎ 〉 ∣ ( 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ∧ 𝜃 ) } |
34 |
|
opabresex2 |
⊢ { 〈 𝑓 , ℎ 〉 ∣ ( 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ∧ 𝜃 ) } ∈ V |
35 |
33 34
|
eqeltri |
⊢ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ∈ V |
36 |
30 31 35
|
ovmpoa |
⊢ ( ( 𝑋 ∈ ( 𝐴 ‘ 𝐺 ) ∧ 𝑌 ∈ ( 𝐵 ‘ 𝐺 ) ) → ( 𝑋 ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) 𝑌 ) = { 〈 𝑓 , ℎ 〉 ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) |
37 |
2 3 36
|
syl2anc |
⊢ ( 𝜑 → ( 𝑋 ( 𝑎 ∈ ( 𝐴 ‘ 𝐺 ) , 𝑏 ∈ ( 𝐵 ‘ 𝐺 ) ↦ { 〈 𝑓 , ℎ 〉 ∣ ( 𝜏 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) 𝑌 ) = { 〈 𝑓 , ℎ 〉 ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) |
38 |
28 37
|
eqtrd |
⊢ ( 𝜑 → ( 𝑋 ( 𝑀 ‘ 𝐺 ) 𝑌 ) = { 〈 𝑓 , ℎ 〉 ∣ ( 𝜃 ∧ 𝑓 ( 𝐷 ‘ 𝐺 ) ℎ ) } ) |