Theorem termcpropd 49862

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 49801	. . 3 ⊢ (𝜑 → (𝐶 ∈ ThinCat ↔ 𝐷 ∈ ThinCat))
6	1	homfeqbas 17631	. . . . 5 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
7	6	eqeq1d 2739	. . . 4 ⊢ (𝜑 → ((Base‘𝐶) = {𝑥} ↔ (Base‘𝐷) = {𝑥}))
8	7	exbidv 1923	. . 3 ⊢ (𝜑 → (∃𝑥(Base‘𝐶) = {𝑥} ↔ ∃𝑥(Base‘𝐷) = {𝑥}))
9	5, 8	anbi12d 633	. 2 ⊢ (𝜑 → ((𝐶 ∈ ThinCat ∧ ∃𝑥(Base‘𝐶) = {𝑥}) ↔ (𝐷 ∈ ThinCat ∧ ∃𝑥(Base‘𝐷) = {𝑥})))
10		eqid 2737	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
11	10	istermc 49833	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥(Base‘𝐶) = {𝑥}))
12		eqid 2737	. . 3 ⊢ (Base‘𝐷) = (Base‘𝐷)
13	12	istermc 49833	. 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 1542  ∃wex 1781   ∈ wcel 2114  {csn 4582  ‘cfv 6500  Basecbs 17148  Hom_f chomf 17601  comp_fccomf 17602  ThinCatcthinc 49776  TermCatctermc 49831

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-cat 17603  df-homf 17605  df-comf 17606  df-thinc 49777  df-termc 49832

This theorem is referenced by:  oppcterm  49865