Theorem thincmo2 49537

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 49530	. . . 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 49536	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  ∃*wmo 2533  ∀wral 3047  ‘cfv 6481  (class class class)co 7346  Basecbs 17120  Hom chom 17172  Catccat 17570  ThinCatcthinc 49528

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-thinc 49529

This theorem is referenced by:  thinchom  49538  thincmo  49539  thincid  49543  thincmon  49544  thincepi  49545  oppcthinco  49550  oppcthinendcALT  49552  functhinclem4  49558  termchommo  49596  funcsn  49652