Theorem istermc 49719

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 6843	. . . 4 ⊢ (𝑐 = 𝐶 → ((Base‘𝑐) = {𝑥} ↔ (Base‘𝐶) = {𝑥}))
2	1	exbidv 1922	. . 3 ⊢ (𝑐 = 𝐶 → (∃𝑥(Base‘𝑐) = {𝑥} ↔ ∃𝑥(Base‘𝐶) = {𝑥}))
3		istermc.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
4	3	eqeq1i 2741	. . . 4 ⊢ (𝐵 = {𝑥} ↔ (Base‘𝐶) = {𝑥})
5	4	exbii 1849	. . 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 49718	. 2 ⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
8	6, 7	elrab2 3649	1 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {csn 4580  ‘cfv 6492  Basecbs 17136  ThinCatcthinc 49662  TermCatctermc 49717

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 2115  ax-9 2123  ax-ext 2708

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-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-termc 49718

This theorem is referenced by:  istermc2  49720  istermc3  49721  termcthin  49722  termcbas  49725  termcpropd  49748  idfudiag1  49770  funcsn  49786  0fucterm  49788  discsnterm  49819