Theorem mapdsn2

Description: Value of the map defined by df-mapd at the span of a singleton. (Contributed by NM, 16-Feb-2015)

	Ref	Expression
Hypotheses	mapdsn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdsn.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
	mapdsn.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdsn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdsn.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
	mapdsn.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
	mapdsn.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
	mapdsn.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
	mapdsn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdsn.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	mapdsn2.e	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( 𝑂 ‘ { 𝑋 } ) )
Assertion	mapdsn2	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = { 𝑓 ∈ 𝐹 ∣ ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ 𝑓 ) } )

Proof

Step	Hyp	Ref	Expression
1		mapdsn.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdsn.o	⊢ 𝑂 = ( ( ocH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdsn.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
4		mapdsn.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		mapdsn.v	⊢ 𝑉 = ( Base ‘ 𝑈 )
6		mapdsn.n	⊢ 𝑁 = ( LSpan ‘ 𝑈 )
7		mapdsn.f	⊢ 𝐹 = ( LFnl ‘ 𝑈 )
8		mapdsn.l	⊢ 𝐿 = ( LKer ‘ 𝑈 )
9		mapdsn.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10		mapdsn.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
11		mapdsn2.e	⊢ ( 𝜑 → ( 𝐿 ‘ 𝐺 ) = ( 𝑂 ‘ { 𝑋 } ) )
12	1 2 3 4 5 6 7 8 9 10	mapdsn	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = { 𝑓 ∈ 𝐹 ∣ ( 𝑂 ‘ { 𝑋 } ) ⊆ ( 𝐿 ‘ 𝑓 ) } )
13	11	sseq1d	⊢ ( 𝜑 → ( ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ 𝑓 ) ↔ ( 𝑂 ‘ { 𝑋 } ) ⊆ ( 𝐿 ‘ 𝑓 ) ) )
14	13	rabbidv	⊢ ( 𝜑 → { 𝑓 ∈ 𝐹 ∣ ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ 𝑓 ) } = { 𝑓 ∈ 𝐹 ∣ ( 𝑂 ‘ { 𝑋 } ) ⊆ ( 𝐿 ‘ 𝑓 ) } )
15	12 14	eqtr4d	⊢ ( 𝜑 → ( 𝑀 ‘ ( 𝑁 ‘ { 𝑋 } ) ) = { 𝑓 ∈ 𝐹 ∣ ( 𝐿 ‘ 𝐺 ) ⊆ ( 𝐿 ‘ 𝑓 ) } )