Step |
Hyp |
Ref |
Expression |
1 |
|
issubc.h |
⊢ 𝐻 = ( Homf ‘ 𝐶 ) |
2 |
|
issubc.i |
⊢ 1 = ( Id ‘ 𝐶 ) |
3 |
|
issubc.o |
⊢ · = ( comp ‘ 𝐶 ) |
4 |
|
issubc.c |
⊢ ( 𝜑 → 𝐶 ∈ Cat ) |
5 |
|
issubc.s |
⊢ ( 𝜑 → 𝑆 = dom dom 𝐽 ) |
6 |
|
simpl |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) → 𝐶 ∈ Cat ) |
7 |
|
sscpwex |
⊢ { 𝑗 ∣ 𝑗 ⊆cat ( Homf ‘ 𝑐 ) } ∈ V |
8 |
|
simpl |
⊢ ( ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) → 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ) |
9 |
8
|
ss2abi |
⊢ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ⊆ { 𝑗 ∣ 𝑗 ⊆cat ( Homf ‘ 𝑐 ) } |
10 |
7 9
|
ssexi |
⊢ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ∈ V |
11 |
10
|
csbex |
⊢ ⦋ 𝐶 / 𝑐 ⦌ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ∈ V |
12 |
11
|
a1i |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) → ⦋ 𝐶 / 𝑐 ⦌ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ∈ V ) |
13 |
|
df-subc |
⊢ Subcat = ( 𝑐 ∈ Cat ↦ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ) |
14 |
13
|
fvmpts |
⊢ ( ( 𝐶 ∈ Cat ∧ ⦋ 𝐶 / 𝑐 ⦌ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ∈ V ) → ( Subcat ‘ 𝐶 ) = ⦋ 𝐶 / 𝑐 ⦌ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ) |
15 |
6 12 14
|
syl2anc |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) → ( Subcat ‘ 𝐶 ) = ⦋ 𝐶 / 𝑐 ⦌ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ) |
16 |
15
|
eleq2d |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) → ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) ↔ 𝐽 ∈ ⦋ 𝐶 / 𝑐 ⦌ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ) ) |
17 |
|
sbcel2 |
⊢ ( [ 𝐶 / 𝑐 ] 𝐽 ∈ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ↔ 𝐽 ∈ ⦋ 𝐶 / 𝑐 ⦌ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ) |
18 |
17
|
a1i |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) → ( [ 𝐶 / 𝑐 ] 𝐽 ∈ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ↔ 𝐽 ∈ ⦋ 𝐶 / 𝑐 ⦌ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ) ) |
19 |
|
elex |
⊢ ( 𝐽 ∈ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } → 𝐽 ∈ V ) |
20 |
19
|
a1i |
⊢ ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) → ( 𝐽 ∈ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } → 𝐽 ∈ V ) ) |
21 |
|
sscrel |
⊢ Rel ⊆cat |
22 |
21
|
brrelex1i |
⊢ ( 𝐽 ⊆cat 𝐻 → 𝐽 ∈ V ) |
23 |
22
|
adantr |
⊢ ( ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) → 𝐽 ∈ V ) |
24 |
23
|
a1i |
⊢ ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) → ( ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) → 𝐽 ∈ V ) ) |
25 |
|
df-sbc |
⊢ ( [ 𝐽 / 𝑗 ] ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) ↔ 𝐽 ∈ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ) |
26 |
|
simpr |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝐽 ∈ V ) → 𝐽 ∈ V ) |
27 |
|
simpr |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) → 𝑗 = 𝐽 ) |
28 |
|
simpr |
⊢ ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) → 𝑐 = 𝐶 ) |
29 |
28
|
fveq2d |
⊢ ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) → ( Homf ‘ 𝑐 ) = ( Homf ‘ 𝐶 ) ) |
30 |
29 1
|
eqtr4di |
⊢ ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) → ( Homf ‘ 𝑐 ) = 𝐻 ) |
31 |
30
|
adantr |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) → ( Homf ‘ 𝑐 ) = 𝐻 ) |
32 |
27 31
|
breq12d |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) → ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ↔ 𝐽 ⊆cat 𝐻 ) ) |
33 |
|
vex |
⊢ 𝑗 ∈ V |
34 |
33
|
dmex |
⊢ dom 𝑗 ∈ V |
35 |
34
|
dmex |
⊢ dom dom 𝑗 ∈ V |
36 |
35
|
a1i |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) → dom dom 𝑗 ∈ V ) |
37 |
27
|
dmeqd |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) → dom 𝑗 = dom 𝐽 ) |
38 |
37
|
dmeqd |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) → dom dom 𝑗 = dom dom 𝐽 ) |
39 |
|
simpllr |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) → 𝑆 = dom dom 𝐽 ) |
40 |
38 39
|
eqtr4d |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) → dom dom 𝑗 = 𝑆 ) |
41 |
|
simpr |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → 𝑠 = 𝑆 ) |
42 |
|
simpllr |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → 𝑐 = 𝐶 ) |
43 |
42
|
fveq2d |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( Id ‘ 𝑐 ) = ( Id ‘ 𝐶 ) ) |
44 |
43 2
|
eqtr4di |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( Id ‘ 𝑐 ) = 1 ) |
45 |
44
|
fveq1d |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) = ( 1 ‘ 𝑥 ) ) |
46 |
|
simplr |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → 𝑗 = 𝐽 ) |
47 |
46
|
oveqd |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( 𝑥 𝑗 𝑥 ) = ( 𝑥 𝐽 𝑥 ) ) |
48 |
45 47
|
eleq12d |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ↔ ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ) ) |
49 |
46
|
oveqd |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( 𝑥 𝑗 𝑦 ) = ( 𝑥 𝐽 𝑦 ) ) |
50 |
46
|
oveqd |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( 𝑦 𝑗 𝑧 ) = ( 𝑦 𝐽 𝑧 ) ) |
51 |
42
|
fveq2d |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( comp ‘ 𝑐 ) = ( comp ‘ 𝐶 ) ) |
52 |
51 3
|
eqtr4di |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( comp ‘ 𝑐 ) = · ) |
53 |
52
|
oveqd |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) = ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) ) |
54 |
53
|
oveqd |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) = ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ) |
55 |
46
|
oveqd |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( 𝑥 𝑗 𝑧 ) = ( 𝑥 𝐽 𝑧 ) ) |
56 |
54 55
|
eleq12d |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ↔ ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) |
57 |
50 56
|
raleqbidv |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ↔ ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) |
58 |
49 57
|
raleqbidv |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ↔ ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) |
59 |
41 58
|
raleqbidv |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ↔ ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) |
60 |
41 59
|
raleqbidv |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ↔ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) |
61 |
48 60
|
anbi12d |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ↔ ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) |
62 |
41 61
|
raleqbidv |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) ∧ 𝑠 = 𝑆 ) → ( ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ↔ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) |
63 |
36 40 62
|
sbcied2 |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) → ( [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ↔ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) |
64 |
32 63
|
anbi12d |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝑗 = 𝐽 ) → ( ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) ↔ ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) ) |
65 |
64
|
adantlr |
⊢ ( ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝐽 ∈ V ) ∧ 𝑗 = 𝐽 ) → ( ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) ↔ ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) ) |
66 |
26 65
|
sbcied |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝐽 ∈ V ) → ( [ 𝐽 / 𝑗 ] ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) ↔ ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) ) |
67 |
25 66
|
bitr3id |
⊢ ( ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) ∧ 𝐽 ∈ V ) → ( 𝐽 ∈ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ↔ ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) ) |
68 |
67
|
ex |
⊢ ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) → ( 𝐽 ∈ V → ( 𝐽 ∈ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ↔ ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) ) ) |
69 |
20 24 68
|
pm5.21ndd |
⊢ ( ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) ∧ 𝑐 = 𝐶 ) → ( 𝐽 ∈ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ↔ ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) ) |
70 |
6 69
|
sbcied |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) → ( [ 𝐶 / 𝑐 ] 𝐽 ∈ { 𝑗 ∣ ( 𝑗 ⊆cat ( Homf ‘ 𝑐 ) ∧ [ dom dom 𝑗 / 𝑠 ] ∀ 𝑥 ∈ 𝑠 ( ( ( Id ‘ 𝑐 ) ‘ 𝑥 ) ∈ ( 𝑥 𝑗 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑠 ∀ 𝑧 ∈ 𝑠 ∀ 𝑓 ∈ ( 𝑥 𝑗 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝑗 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 ( comp ‘ 𝑐 ) 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝑗 𝑧 ) ) ) } ↔ ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) ) |
71 |
16 18 70
|
3bitr2d |
⊢ ( ( 𝐶 ∈ Cat ∧ 𝑆 = dom dom 𝐽 ) → ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) ↔ ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) ) |
72 |
4 5 71
|
syl2anc |
⊢ ( 𝜑 → ( 𝐽 ∈ ( Subcat ‘ 𝐶 ) ↔ ( 𝐽 ⊆cat 𝐻 ∧ ∀ 𝑥 ∈ 𝑆 ( ( 1 ‘ 𝑥 ) ∈ ( 𝑥 𝐽 𝑥 ) ∧ ∀ 𝑦 ∈ 𝑆 ∀ 𝑧 ∈ 𝑆 ∀ 𝑓 ∈ ( 𝑥 𝐽 𝑦 ) ∀ 𝑔 ∈ ( 𝑦 𝐽 𝑧 ) ( 𝑔 ( 〈 𝑥 , 𝑦 〉 · 𝑧 ) 𝑓 ) ∈ ( 𝑥 𝐽 𝑧 ) ) ) ) ) |