Theorem termcpropd 49108

Description: Two structures with the same base, hom-sets and composition operation are either both terminal categories or neither. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypotheses

Ref	Expression
termcpropd.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
termcpropd.2	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
termcpropd.3	⊢ (𝜑 → 𝐶 ∈ 𝑉)
termcpropd.4	⊢ (𝜑 → 𝐷 ∈ 𝑊)

Assertion

Ref	Expression
termcpropd	⊢ (𝜑 → (𝐶 ∈ TermCat ↔ 𝐷 ∈ TermCat))

Proof of Theorem termcpropd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termcpropd.1	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
2		termcpropd.2	. . . 4 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
3		termcpropd.3	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
4		termcpropd.4	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
5	1, 2, 3, 4	thincpropd 49064	. . 3 ⊢ (𝜑 → (𝐶 ∈ ThinCat ↔ 𝐷 ∈ ThinCat))
6	1	homfeqbas 17735	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
7	6	eqeq1d 2738	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) = {𝑥} ↔ (Base‘𝐷) = {𝑥}))
8	7	exbidv 1921	. . 3 ⊢ (𝜑 → (∃𝑥(Base‘𝐶) = {𝑥} ↔ ∃𝑥(Base‘𝐷) = {𝑥}))
9	5, 8	anbi12d 632	. 2 ⊢ (𝜑 → ((𝐶 ∈ ThinCat ∧ ∃𝑥(Base‘𝐶) = {𝑥}) ↔ (𝐷 ∈ ThinCat ∧ ∃𝑥(Base‘𝐷) = {𝑥})))
10		eqid 2736	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
11	10	istermc 49094	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥(Base‘𝐶) = {𝑥}))
12		eqid 2736	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
13	12	istermc 49094	. 2 ⊢ (𝐷 ∈ TermCat ↔ (𝐷 ∈ ThinCat ∧ ∃𝑥(Base‘𝐷) = {𝑥}))
14	9, 11, 13	3bitr4g 314	1 ⊢ (𝜑 → (𝐶 ∈ TermCat ↔ 𝐷 ∈ TermCat))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {csn 4624  ‘cfv 6559  Basecbs 17243  Hom_f chomf 17705  comp_fccomf 17706  ThinCatcthinc 49040  TermCatctermc 49092

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5277  ax-sep 5294  ax-nul 5304  ax-pow 5363  ax-pr 5430  ax-un 7751

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-1st 8010  df-2nd 8011  df-cat 17707  df-homf 17709  df-comf 17710  df-thinc 49041  df-termc 49093

This theorem is referenced by:  oppcterm  49111