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Metamath Proof Explorer

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 ) )