Theorem thincmoALT

Description: Alternate proof of thincmo . (Contributed by Zhi Wang, 21-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	thincmo.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
	thincmo.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	thincmo.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	thincmo.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	thincmo.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
Assertion	thincmoALT	⊢ ( 𝜑 → ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		thincmo.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
2		thincmo.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
3		thincmo.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
4		thincmo.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
5		thincmo.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
6	4 5	isthinc	⊢ ( 𝐶 ∈ ThinCat ↔ ( 𝐶 ∈ Cat ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∃* 𝑓 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ) )
7	6	simprbi	⊢ ( 𝐶 ∈ ThinCat → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∃* 𝑓 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) )
8	1 7	syl	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∃* 𝑓 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) )
9		oveq12	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑥 𝐻 𝑦 ) = ( 𝑋 𝐻 𝑌 ) )
10	9	eleq2d	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ↔ 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) )
11	10	mobidv	⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( ∃* 𝑓 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) ↔ ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) )
12	11	rspc2gv	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∃* 𝑓 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) → ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) )
13	2 3 12	syl2anc	⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ∃* 𝑓 𝑓 ∈ ( 𝑥 𝐻 𝑦 ) → ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) ) )
14	8 13	mpd	⊢ ( 𝜑 → ∃* 𝑓 𝑓 ∈ ( 𝑋 𝐻 𝑌 ) )