Theorem lindsmm2

Description: The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015)

	Ref	Expression
Hypotheses	lindfmm.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
	lindfmm.c	⊢ 𝐶 = ( Base ‘ 𝑇 )
Assertion	lindsmm2	⊢ ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → ( 𝐺 “ 𝐹 ) ∈ ( LIndS ‘ 𝑇 ) )

Proof

Step	Hyp	Ref	Expression
1		lindfmm.b	⊢ 𝐵 = ( Base ‘ 𝑆 )
2		lindfmm.c	⊢ 𝐶 = ( Base ‘ 𝑇 )
3		simp3	⊢ ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → 𝐹 ∈ ( LIndS ‘ 𝑆 ) )
4	1	linds1	⊢ ( 𝐹 ∈ ( LIndS ‘ 𝑆 ) → 𝐹 ⊆ 𝐵 )
5	1 2	lindsmm	⊢ ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 ⊆ 𝐵 ) → ( 𝐹 ∈ ( LIndS ‘ 𝑆 ) ↔ ( 𝐺 “ 𝐹 ) ∈ ( LIndS ‘ 𝑇 ) ) )
6	4 5	syl3an3	⊢ ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → ( 𝐹 ∈ ( LIndS ‘ 𝑆 ) ↔ ( 𝐺 “ 𝐹 ) ∈ ( LIndS ‘ 𝑇 ) ) )
7	3 6	mpbid	⊢ ( ( 𝐺 ∈ ( 𝑆 LMHom 𝑇 ) ∧ 𝐺 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 ∈ ( LIndS ‘ 𝑆 ) ) → ( 𝐺 “ 𝐹 ) ∈ ( LIndS ‘ 𝑇 ) )