Theorem moni

Description: Property of a monomorphism. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	ismon.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	ismon.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	ismon.o	⊢ · = ( comp ‘ 𝐶 )
	ismon.s	⊢ 𝑀 = ( Mono ‘ 𝐶 )
	ismon.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
	ismon.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	ismon.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	moni.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
	moni.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝑀 𝑌 ) )
	moni.g	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑍 𝐻 𝑋 ) )
	moni.k	⊢ ( 𝜑 → 𝐾 ∈ ( 𝑍 𝐻 𝑋 ) )
Assertion	moni	⊢ ( 𝜑 → ( ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐺 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐾 ) ↔ 𝐺 = 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		ismon.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
2		ismon.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
3		ismon.o	⊢ · = ( comp ‘ 𝐶 )
4		ismon.s	⊢ 𝑀 = ( Mono ‘ 𝐶 )
5		ismon.c	⊢ ( 𝜑 → 𝐶 ∈ Cat )
6		ismon.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
7		ismon.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
8		moni.z	⊢ ( 𝜑 → 𝑍 ∈ 𝐵 )
9		moni.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝑀 𝑌 ) )
10		moni.g	⊢ ( 𝜑 → 𝐺 ∈ ( 𝑍 𝐻 𝑋 ) )
11		moni.k	⊢ ( 𝜑 → 𝐾 ∈ ( 𝑍 𝐻 𝑋 ) )
12	1 2 3 4 5 6 7	ismon2	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝑀 𝑌 ) ↔ ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ∀ 𝑔 ∈ ( 𝑧 𝐻 𝑋 ) ∀ ℎ ∈ ( 𝑧 𝐻 𝑋 ) ( ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) 𝑔 ) = ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) ℎ ) → 𝑔 = ℎ ) ) ) )
13	9 12	mpbid	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ∀ 𝑔 ∈ ( 𝑧 𝐻 𝑋 ) ∀ ℎ ∈ ( 𝑧 𝐻 𝑋 ) ( ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) 𝑔 ) = ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) ℎ ) → 𝑔 = ℎ ) ) )
14	13	simprd	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐵 ∀ 𝑔 ∈ ( 𝑧 𝐻 𝑋 ) ∀ ℎ ∈ ( 𝑧 𝐻 𝑋 ) ( ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) 𝑔 ) = ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) ℎ ) → 𝑔 = ℎ ) )
15	10	adantr	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → 𝐺 ∈ ( 𝑍 𝐻 𝑋 ) )
16		simpr	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → 𝑧 = 𝑍 )
17	16	oveq1d	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → ( 𝑧 𝐻 𝑋 ) = ( 𝑍 𝐻 𝑋 ) )
18	15 17	eleqtrrd	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → 𝐺 ∈ ( 𝑧 𝐻 𝑋 ) )
19	11	adantr	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → 𝐾 ∈ ( 𝑍 𝐻 𝑋 ) )
20	19 17	eleqtrrd	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → 𝐾 ∈ ( 𝑧 𝐻 𝑋 ) )
21	20	adantr	⊢ ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) → 𝐾 ∈ ( 𝑧 𝐻 𝑋 ) )
22		simpllr	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → 𝑧 = 𝑍 )
23	22	opeq1d	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → ⟨ 𝑧 , 𝑋 ⟩ = ⟨ 𝑍 , 𝑋 ⟩ )
24	23	oveq1d	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) = ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) )
25		eqidd	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → 𝐹 = 𝐹 )
26		simplr	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → 𝑔 = 𝐺 )
27	24 25 26	oveq123d	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) 𝑔 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐺 ) )
28		simpr	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → ℎ = 𝐾 )
29	24 25 28	oveq123d	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) ℎ ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐾 ) )
30	27 29	eqeq12d	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → ( ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) 𝑔 ) = ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) ℎ ) ↔ ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐺 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐾 ) ) )
31	26 28	eqeq12d	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → ( 𝑔 = ℎ ↔ 𝐺 = 𝐾 ) )
32	30 31	imbi12d	⊢ ( ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) ∧ ℎ = 𝐾 ) → ( ( ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) 𝑔 ) = ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) ℎ ) → 𝑔 = ℎ ) ↔ ( ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐺 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐾 ) → 𝐺 = 𝐾 ) ) )
33	21 32	rspcdv	⊢ ( ( ( 𝜑 ∧ 𝑧 = 𝑍 ) ∧ 𝑔 = 𝐺 ) → ( ∀ ℎ ∈ ( 𝑧 𝐻 𝑋 ) ( ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) 𝑔 ) = ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) ℎ ) → 𝑔 = ℎ ) → ( ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐺 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐾 ) → 𝐺 = 𝐾 ) ) )
34	18 33	rspcimdv	⊢ ( ( 𝜑 ∧ 𝑧 = 𝑍 ) → ( ∀ 𝑔 ∈ ( 𝑧 𝐻 𝑋 ) ∀ ℎ ∈ ( 𝑧 𝐻 𝑋 ) ( ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) 𝑔 ) = ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) ℎ ) → 𝑔 = ℎ ) → ( ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐺 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐾 ) → 𝐺 = 𝐾 ) ) )
35	8 34	rspcimdv	⊢ ( 𝜑 → ( ∀ 𝑧 ∈ 𝐵 ∀ 𝑔 ∈ ( 𝑧 𝐻 𝑋 ) ∀ ℎ ∈ ( 𝑧 𝐻 𝑋 ) ( ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) 𝑔 ) = ( 𝐹 ( ⟨ 𝑧 , 𝑋 ⟩ · 𝑌 ) ℎ ) → 𝑔 = ℎ ) → ( ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐺 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐾 ) → 𝐺 = 𝐾 ) ) )
36	14 35	mpd	⊢ ( 𝜑 → ( ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐺 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐾 ) → 𝐺 = 𝐾 ) )
37		oveq2	⊢ ( 𝐺 = 𝐾 → ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐺 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐾 ) )
38	36 37	impbid1	⊢ ( 𝜑 → ( ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐺 ) = ( 𝐹 ( ⟨ 𝑍 , 𝑋 ⟩ · 𝑌 ) 𝐾 ) ↔ 𝐺 = 𝐾 ) )