Theorem istermc 49479

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 6835	. . . 4 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) = {𝑥} ↔ (Base‘𝐶) = {𝑥}))
2	1	exbidv 1921	. . 3 ⊢ (𝑐 = 𝐶 → (∃𝑥(Base‘𝑐) = {𝑥} ↔ ∃𝑥(Base‘𝐶) = {𝑥}))
3		istermc.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
4	3	eqeq1i 2734	. . . 4 ⊢ (𝐵 = {𝑥} ↔ (Base‘𝐶) = {𝑥})
5	4	exbii 1848	. . 3 ⊢ (∃𝑥 𝐵 = {𝑥} ↔ ∃𝑥(Base‘𝐶) = {𝑥})
6	2, 5	bitr4di 289	. 2 ⊢ (𝑐 = 𝐶 → (∃𝑥(Base‘𝑐) = {𝑥} ↔ ∃𝑥 𝐵 = {𝑥}))
7		df-termc 49478	. 2 ⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
8	6, 7	elrab2 3653	1 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {csn 4579  ‘cfv 6486  Basecbs 17139  ThinCatcthinc 49422  TermCatctermc 49477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-termc 49478

This theorem is referenced by:  istermc2  49480  istermc3  49481  termcthin  49482  termcbas  49485  termcpropd  49508  idfudiag1  49530  funcsn  49546  0fucterm  49548  discsnterm  49579