Metamath Proof Explorer


Theorem coafval

Description: The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypotheses coafval.o · ˙ = comp a C
coafval.a A = Arrow C
coafval.x ˙ = comp C
Assertion coafval · ˙ = g A , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f

Proof

Step Hyp Ref Expression
1 coafval.o · ˙ = comp a C
2 coafval.a A = Arrow C
3 coafval.x ˙ = comp C
4 fveq2 c = C Arrow c = Arrow C
5 4 2 eqtr4di c = C Arrow c = A
6 5 rabeqdv c = C h Arrow c | cod a h = dom a g = h A | cod a h = dom a g
7 fveq2 c = C comp c = comp C
8 7 3 eqtr4di c = C comp c = ˙
9 8 oveqd c = C dom a f dom a g comp c cod a g = dom a f dom a g ˙ cod a g
10 9 oveqd c = C 2 nd g dom a f dom a g comp c cod a g 2 nd f = 2 nd g dom a f dom a g ˙ cod a g 2 nd f
11 10 oteq3d c = C dom a f cod a g 2 nd g dom a f dom a g comp c cod a g 2 nd f = dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f
12 5 6 11 mpoeq123dv c = C g Arrow c , f h Arrow c | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g comp c cod a g 2 nd f = g A , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f
13 df-coa comp a = c Cat g Arrow c , f h Arrow c | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g comp c cod a g 2 nd f
14 2 fvexi A V
15 14 rabex h A | cod a h = dom a g V
16 14 15 mpoex g A , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f V
17 12 13 16 fvmpt C Cat comp a C = g A , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f
18 13 fvmptndm ¬ C Cat comp a C =
19 2 arwrcl f A C Cat
20 19 con3i ¬ C Cat ¬ f A
21 20 eq0rdv ¬ C Cat A =
22 eqidd ¬ C Cat h A | cod a h = dom a g = h A | cod a h = dom a g
23 eqidd ¬ C Cat dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f = dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f
24 21 22 23 mpoeq123dv ¬ C Cat g A , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f = g , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f
25 mpo0 g , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f =
26 24 25 eqtrdi ¬ C Cat g A , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f =
27 18 26 eqtr4d ¬ C Cat comp a C = g A , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f
28 17 27 pm2.61i comp a C = g A , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f
29 1 28 eqtri · ˙ = g A , f h A | cod a h = dom a g dom a f cod a g 2 nd g dom a f dom a g ˙ cod a g 2 nd f