Step |
Hyp |
Ref |
Expression |
1 |
|
ismon.b |
⊢ 𝐵 = ( Base ‘ 𝐶 ) |
2 |
|
ismon.h |
⊢ 𝐻 = ( Hom ‘ 𝐶 ) |
3 |
|
ismon.o |
⊢ · = ( comp ‘ 𝐶 ) |
4 |
|
ismon.s |
⊢ 𝑀 = ( Mono ‘ 𝐶 ) |
5 |
|
ismon.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
6 |
|
fvexd |
⊢ ( 𝑐 = 𝐶 → ( Base ‘ 𝑐 ) ∈ V ) |
7 |
|
fveq2 |
⊢ ( 𝑐 = 𝐶 → ( Base ‘ 𝑐 ) = ( Base ‘ 𝐶 ) ) |
8 |
7 1
|
eqtr4di |
⊢ ( 𝑐 = 𝐶 → ( Base ‘ 𝑐 ) = 𝐵 ) |
9 |
|
fvexd |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑐 ) ∈ V ) |
10 |
|
simpl |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) → 𝑐 = 𝐶 ) |
11 |
10
|
fveq2d |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑐 ) = ( Hom ‘ 𝐶 ) ) |
12 |
11 2
|
eqtr4di |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) → ( Hom ‘ 𝑐 ) = 𝐻 ) |
13 |
|
simplr |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → 𝑏 = 𝐵 ) |
14 |
|
simpr |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ℎ = 𝐻 ) |
15 |
14
|
oveqd |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( 𝑥 ℎ 𝑦 ) = ( 𝑥 𝐻 𝑦 ) ) |
16 |
14
|
oveqd |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( 𝑧 ℎ 𝑥 ) = ( 𝑧 𝐻 𝑥 ) ) |
17 |
|
simpll |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → 𝑐 = 𝐶 ) |
18 |
17
|
fveq2d |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( comp ‘ 𝑐 ) = ( comp ‘ 𝐶 ) ) |
19 |
18 3
|
eqtr4di |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( comp ‘ 𝑐 ) = · ) |
20 |
19
|
oveqd |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) = ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) ) |
21 |
20
|
oveqd |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( 𝑓 ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) 𝑔 ) = ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) |
22 |
16 21
|
mpteq12dv |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( 𝑔 ∈ ( 𝑧 ℎ 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) 𝑔 ) ) = ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) ) |
23 |
22
|
cnveqd |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ◡ ( 𝑔 ∈ ( 𝑧 ℎ 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) 𝑔 ) ) = ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) ) |
24 |
23
|
funeqd |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( Fun ◡ ( 𝑔 ∈ ( 𝑧 ℎ 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) 𝑔 ) ) ↔ Fun ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) ) ) |
25 |
13 24
|
raleqbidv |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( ∀ 𝑧 ∈ 𝑏 Fun ◡ ( 𝑔 ∈ ( 𝑧 ℎ 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) 𝑔 ) ) ↔ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) ) ) |
26 |
15 25
|
rabeqbidv |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → { 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ∣ ∀ 𝑧 ∈ 𝑏 Fun ◡ ( 𝑔 ∈ ( 𝑧 ℎ 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) 𝑔 ) ) } = { 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∣ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) } ) |
27 |
13 13 26
|
mpoeq123dv |
⊢ ( ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) ∧ ℎ = 𝐻 ) → ( 𝑥 ∈ 𝑏 , 𝑦 ∈ 𝑏 ↦ { 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ∣ ∀ 𝑧 ∈ 𝑏 Fun ◡ ( 𝑔 ∈ ( 𝑧 ℎ 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) 𝑔 ) ) } ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∣ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) } ) ) |
28 |
9 12 27
|
csbied2 |
⊢ ( ( 𝑐 = 𝐶 ∧ 𝑏 = 𝐵 ) → ⦋ ( Hom ‘ 𝑐 ) / ℎ ⦌ ( 𝑥 ∈ 𝑏 , 𝑦 ∈ 𝑏 ↦ { 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ∣ ∀ 𝑧 ∈ 𝑏 Fun ◡ ( 𝑔 ∈ ( 𝑧 ℎ 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) 𝑔 ) ) } ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∣ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) } ) ) |
29 |
6 8 28
|
csbied2 |
⊢ ( 𝑐 = 𝐶 → ⦋ ( Base ‘ 𝑐 ) / 𝑏 ⦌ ⦋ ( Hom ‘ 𝑐 ) / ℎ ⦌ ( 𝑥 ∈ 𝑏 , 𝑦 ∈ 𝑏 ↦ { 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ∣ ∀ 𝑧 ∈ 𝑏 Fun ◡ ( 𝑔 ∈ ( 𝑧 ℎ 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) 𝑔 ) ) } ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∣ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) } ) ) |
30 |
|
df-mon |
⊢ Mono = ( 𝑐 ∈ Cat ↦ ⦋ ( Base ‘ 𝑐 ) / 𝑏 ⦌ ⦋ ( Hom ‘ 𝑐 ) / ℎ ⦌ ( 𝑥 ∈ 𝑏 , 𝑦 ∈ 𝑏 ↦ { 𝑓 ∈ ( 𝑥 ℎ 𝑦 ) ∣ ∀ 𝑧 ∈ 𝑏 Fun ◡ ( 𝑔 ∈ ( 𝑧 ℎ 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 ( comp ‘ 𝑐 ) 𝑦 ) 𝑔 ) ) } ) ) |
31 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
32 |
31 31
|
mpoex |
⊢ ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∣ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) } ) ∈ V |
33 |
29 30 32
|
fvmpt |
⊢ ( 𝐶 ∈ Cat → ( Mono ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∣ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) } ) ) |
34 |
5 33
|
syl |
⊢ ( 𝜑 → ( Mono ‘ 𝐶 ) = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∣ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) } ) ) |
35 |
4 34
|
eqtrid |
⊢ ( 𝜑 → 𝑀 = ( 𝑥 ∈ 𝐵 , 𝑦 ∈ 𝐵 ↦ { 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ∣ ∀ 𝑧 ∈ 𝐵 Fun ◡ ( 𝑔 ∈ ( 𝑧 𝐻 𝑥 ) ↦ ( 𝑓 ( 〈 𝑧 , 𝑥 〉 · 𝑦 ) 𝑔 ) ) } ) ) |