Theorem thincmo2 50355

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 50348	. . . 4 ⊢ (𝐶 ∈ ThinCat ↔ (𝐶 ∈ Cat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦)))
9	8	simprbi 503	. . 3 ⊢ (𝐶 ∈ ThinCat → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
10	5, 9	syl 18	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
11	1, 2, 3, 4, 10	isthincd2lem1 50354	1 ⊢ (𝜑 → 𝐹 = 𝐺)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∃*wmo 2562  ∀wral 3076  ‘cfv 6533  (class class class)co 7414  Basecbs 17304  Hom chom 17356  Catccat 17755  ThinCatcthinc 50346

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-nul 5263

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417  df-thinc 50347

This theorem is used by:  thinchom  50356  thincmo  50357  thincid  50361  thincmon  50362  thincepi  50363  oppcthinco  50368  oppcthinendcALT  50370  functhinclem4  50376  termchommo  50414  funcsn  50470