Theorem istermc 50551

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 6892	. . . 4 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) = {𝑥} ↔ (Base‘𝐶) = {𝑥}))
2	1	exbidv 1954	. . 3 ⊢ (𝑐 = 𝐶 → (∃𝑥(Base‘𝑐) = {𝑥} ↔ ∃𝑥(Base‘𝐶) = {𝑥}))
3		istermc.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
4	3	eqeq1i 2766	. . . 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 50550	. 2 ⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
8	6, 7	elrab2 3649	1 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {csn 4584  ‘cfv 6537  Basecbs 17380  ThinCatcthinc 50494  TermCatctermc 50549

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-ext 2733

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 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  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 6493  df-fv 6545  df-termc 50550

This theorem is used by:  istermc2  50552  istermc3  50553  termcthin  50554  termcbas  50557  termcpropd  50580  idfudiag1  50602  funcsn  50618  0fucterm  50620  discsnterm  50651