Theorem thincmo2 47736

Description: Morphisms in the same hom-set are identical. (Contributed by Zhi Wang, 17-Sep-2024.)

Hypotheses

Ref	Expression
isthincd2lem1.1	⊢ (𝜑 → 𝑋 ∈ 𝐵)
isthincd2lem1.2	⊢ (𝜑 → 𝑌 ∈ 𝐵)
isthincd2lem1.3	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
isthincd2lem1.4	⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
thincmo2.b	⊢ 𝐵 = (Base‘𝐶)
thincmo2.h	⊢ 𝐻 = (Hom ‘𝐶)
thincmo2.c	⊢ (𝜑 → 𝐶 ∈ ThinCat)

Assertion

Ref	Expression
thincmo2	⊢ (𝜑 → 𝐹 = 𝐺)

Proof of Theorem thincmo2

Dummy variables 𝑦 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isthincd2lem1.1	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		isthincd2lem1.2	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
3		isthincd2lem1.3	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
4		isthincd2lem1.4	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
5		thincmo2.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ ThinCat)
6		thincmo2.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
7		thincmo2.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
8	6, 7	isthinc 47729	. . . 4 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
9	8	simprbi 496	. . 3 ⊢ (𝐶 ∈ ThinCat → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
10	5, 9	syl 17	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
11	1, 2, 3, 4, 10	isthincd2lem1 47735	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105  ∃*wmo 2531  ∀wral 3060  ‘cfv 6543  (class class class)co 7412  Basecbs 17149  Hom chom 17213  Catccat 17613  ThinCatcthinc 47727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7415  df-thinc 47728

This theorem is referenced by:  thincmo  47737  thincid  47741  thincmon  47742  thincepi  47743  functhinclem4  47752