Theorem istermc 50716

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 6897	. . . 4 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) = {𝑥} ↔ (Base‘𝐶) = {𝑥}))
2	1	exbidv 1953	. . 3 ⊢ (𝑐 = 𝐶 → (∃𝑥(Base‘𝑐) = {𝑥} ↔ ∃𝑥(Base‘𝐶) = {𝑥}))
3		istermc.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
4	3	eqeq1i 2765	. . . 4 ⊢ (𝐵 = {𝑥} ↔ (Base‘𝐶) = {𝑥})
5	4	exbii 1880	. . 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 50715	. 2 ⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
8	6, 7	elrab2 3648	1 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {csn 4584  ‘cfv 6541  Basecbs 17394  ThinCatcthinc 50659  TermCatctermc 50714

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-iota 6496  df-fv 6549  df-termc 50715

This theorem is used by:  istermc2  50717  istermc3  50718  termcthin  50719  termcbas  50722  termcpropd  50745  idfudiag1  50767  funcsn  50786  0fucterm  50788  discsnterm  50819