Theorem termcpropd 49535

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 49474	. . 3 ⊢ (𝜑 → (𝐶 ∈ ThinCat ↔ 𝐷 ∈ ThinCat))
6	1	homfeqbas 17597	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
7	6	eqeq1d 2733	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) = {𝑥} ↔ (Base‘𝐷) = {𝑥}))
8	7	exbidv 1922	. . 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 2731	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
11	10	istermc 49506	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥(Base‘𝐶) = {𝑥}))
12		eqid 2731	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
13	12	istermc 49506	. 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 1541  ∃wex 1780   ∈ wcel 2111  {csn 4571  ‘cfv 6476  Basecbs 17115  Hom_f chomf 17567  comp_fccomf 17568  ThinCatcthinc 49449  TermCatctermc 49504

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

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-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-cat 17569  df-homf 17571  df-comf 17572  df-thinc 49450  df-termc 49505

This theorem is referenced by:  oppcterm  49538