Theorem termcpropd 50969

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 50908	. . 3 ⊢ (𝜑 → (𝐶 ∈ ThinCat ↔ 𝐷 ∈ ThinCat))
6	1	homfeqbas 17920	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
7	6	eqeq1d 2765	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) = {𝑥} ↔ (Base‘𝐷) = {𝑥}))
8	7	exbidv 1956	. . 3 ⊢ (𝜑 → (∃𝑥(Base‘𝐶) = {𝑥} ↔ ∃𝑥(Base‘𝐷) = {𝑥}))
9	5, 8	anbi12d 644	. 2 ⊢ (𝜑 → ((𝐶 ∈ ThinCat ∧ ∃𝑥(Base‘𝐶) = {𝑥}) ↔ (𝐷 ∈ ThinCat ∧ ∃𝑥(Base‘𝐷) = {𝑥})))
10		eqid 2763	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
11	10	istermc 50940	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥(Base‘𝐶) = {𝑥}))
12		eqid 2763	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
13	12	istermc 50940	. 2 ⊢ (𝐷 ∈ TermCat ↔ (𝐷 ∈ ThinCat ∧ ∃𝑥(Base‘𝐷) = {𝑥}))
14	9, 11, 13	3bitr4g 317	1 ⊢ (𝜑 → (𝐶 ∈ TermCat ↔ 𝐷 ∈ TermCat))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {csn 4588  ‘cfv 6566  Basecbs 17431  Hom_f chomf 17890  comp_fccomf 17891  ThinCatcthinc 50883  TermCatctermc 50938

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-cat 17892  df-homf 17894  df-comf 17895  df-thinc 50884  df-termc 50939

This theorem is used by:  oppcterm  50972