Theorem mapd11

Description: The map defined by df-mapd is one-to-one. Property (c) of Baer p. 40. (Contributed by NM, 12-Mar-2015)

	Ref	Expression
Hypotheses	mapdord.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	mapdord.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
	mapdord.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
	mapdord.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
	mapdord.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
	mapdord.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
	mapdord.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
Assertion	mapd11	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		mapdord.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		mapdord.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
3		mapdord.s	⊢ 𝑆 = ( LSubSp ‘ 𝑈 )
4		mapdord.m	⊢ 𝑀 = ( ( mapd ‘ 𝐾 ) ‘ 𝑊 )
5		mapdord.k	⊢ ( 𝜑 → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
6		mapdord.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
7		mapdord.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑆 )
8	1 2 3 4 5 6 7	mapdord	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 ⊆ 𝑌 ) )
9	1 2 3 4 5 7 6	mapdord	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ↔ 𝑌 ⊆ 𝑋 ) )
10	8 9	anbi12d	⊢ ( 𝜑 → ( ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ∧ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) ↔ ( 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋 ) ) )
11		eqss	⊢ ( ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑌 ) ↔ ( ( 𝑀 ‘ 𝑋 ) ⊆ ( 𝑀 ‘ 𝑌 ) ∧ ( 𝑀 ‘ 𝑌 ) ⊆ ( 𝑀 ‘ 𝑋 ) ) )
12		eqss	⊢ ( 𝑋 = 𝑌 ↔ ( 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋 ) )
13	10 11 12	3bitr4g	⊢ ( 𝜑 → ( ( 𝑀 ‘ 𝑋 ) = ( 𝑀 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) )