Theorem mapdhval0

Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015)

	Ref	Expression
Hypotheses	mapdh.q	$$⊢Q=0_C$$
	mapdh.i	$$⊢I=\left(x\in \mathrm{V}⟼if\left(2^{nd}\left(x\right)=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{2^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{1^{st}\left(1^{st}\left(x\right)\right)\underset{\dot{}}{-}{2}^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{2^{nd}\left(1^{st}\left(x\right)\right)Rh\right\}\right)\right)\right)\right)\right)$$
	mapdh0.o	$$⊢\underset{\dot{}}{0}=0_U$$
	mapdh0.x	$$⊢\phi \to X\in A$$
	mapdh0.f	$$⊢\phi \to F\in B$$
Assertion	mapdhval0	$$⊢\phi \to I\left(⟨X,F,\underset{\dot{}}{0}⟩\right)=Q$$

Proof

Step	Hyp	Ref	Expression
1		mapdh.q	$$⊢Q=0_C$$
2		mapdh.i	$$⊢I=\left(x\in \mathrm{V}⟼if\left(2^{nd}\left(x\right)=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{2^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{1^{st}\left(1^{st}\left(x\right)\right)\underset{\dot{}}{-}{2}^{nd}\left(x\right)\right\}\right)\right)=J\left(\left\{2^{nd}\left(1^{st}\left(x\right)\right)Rh\right\}\right)\right)\right)\right)\right)$$
3		mapdh0.o	$$⊢\underset{\dot{}}{0}=0_U$$
4		mapdh0.x	$$⊢\phi \to X\in A$$
5		mapdh0.f	$$⊢\phi \to F\in B$$
6	3	fvexi	$$⊢\underset{\dot{}}{0}\in \mathrm{V}$$
7	6	a1i	$$⊢\phi \to \underset{\dot{}}{0}\in \mathrm{V}$$
8	1 2 4 5 7	mapdhval	$$⊢\phi \to I\left(⟨X,F,\underset{\dot{}}{0}⟩\right)=if\left(\underset{\dot{}}{0}=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{\underset{\dot{}}{0}\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{X\underset{\dot{}}{-}\underset{\dot{}}{0}\right\}\right)\right)=J\left(\left\{FRh\right\}\right)\right)\right)\right)$$
9		eqid	$$⊢\underset{\dot{}}{0}=\underset{\dot{}}{0}$$
10	9	iftruei	$$⊢if\left(\underset{\dot{}}{0}=\underset{\dot{}}{0},Q,\left(\iota h\in D|\left(M\left(N\left(\left\{\underset{\dot{}}{0}\right\}\right)\right)=J\left(\left\{h\right\}\right)\wedge M\left(N\left(\left\{X\underset{\dot{}}{-}\underset{\dot{}}{0}\right\}\right)\right)=J\left(\left\{FRh\right\}\right)\right)\right)\right)=Q$$
11	8 10	eqtrdi	$$⊢\phi \to I\left(⟨X,F,\underset{\dot{}}{0}⟩\right)=Q$$