Theorem istermc 50940

Description: The predicate "is a terminal category". A terminal category is a thin category with a singleton base set. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypothesis

Ref	Expression
istermc.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
istermc	⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))

Distinct variable group:   𝑥,𝐶

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem istermc

Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveqeq2 6925	. . . 4 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) = {𝑥} ↔ (Base‘𝐶) = {𝑥}))
2	1	exbidv 1956	. . 3 ⊢ (𝑐 = 𝐶 → (∃𝑥(Base‘𝑐) = {𝑥} ↔ ∃𝑥(Base‘𝐶) = {𝑥}))
3		istermc.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
4	3	eqeq1i 2768	. . . 4 ⊢ (𝐵 = {𝑥} ↔ (Base‘𝐶) = {𝑥})
5	4	exbii 1881	. . 3 ⊢ (∃𝑥 𝐵 = {𝑥} ↔ ∃𝑥(Base‘𝐶) = {𝑥})
6	2, 5	bitr4di 292	. 2 ⊢ (𝑐 = 𝐶 → (∃𝑥(Base‘𝑐) = {𝑥} ↔ ∃𝑥 𝐵 = {𝑥}))
7		df-termc 50939	. 2 ⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
8	6, 7	elrab2 3652	1 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {csn 4588  ‘cfv 6566  Basecbs 17431  ThinCatcthinc 50883  TermCatctermc 50938

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

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-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-termc 50939

This theorem is used by:  istermc2  50941  istermc3  50942  termcthin  50943  termcbas  50946  termcpropd  50969  idfudiag1  50991  funcsn  51010  0fucterm  51012  discsnterm  51043