Theorem termopropdlem 49743

Description: Lemma for termopropd 49746. (Contributed by Zhi Wang, 26-Oct-2025.)

Hypotheses

Ref	Expression
initopropd.1	⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
initopropd.2	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))
initopropdlem.1	⊢ (𝜑 → ¬ 𝐶 ∈ V)

Assertion

Ref	Expression
termopropdlem	⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))

Proof of Theorem termopropdlem

Dummy variables 𝑎 𝑏 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		termofn 17950	. 2 ⊢ TermO Fn Cat
2		initopropdlem.1	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ V)
3		ssv 3940	. 2 ⊢ Cat ⊆ V
4		simpr 486	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat)
5		eqid 2741	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
6		eqid 2741	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7	4, 5, 6	termoval 17956	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (TermO‘𝐷) = {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)})
8		initopropd.1	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
9		fvprc 6822	. . . . . . . . 9 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅)
10	2, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐶) = ∅)
11	8, 10	eqtr3d 2778	. . . . . . 7 ⊢ (𝜑 → (Homf ‘𝐷) = ∅)
12		homf0 49511	. . . . . . 7 ⊢ ((Base‘𝐷) = ∅ ↔ (Homf ‘𝐷) = ∅)
13	11, 12	sylibr 236	. . . . . 6 ⊢ (𝜑 → (Base‘𝐷) = ∅)
14	13	rabeqdv 3408	. . . . 5 ⊢ (𝜑 → {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)} = {𝑎 ∈ ∅ ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)})
15		rab0 4316	. . . . 5 ⊢ {𝑎 ∈ ∅ ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)} = ∅
16	14, 15	eqtrdi 2792	. . . 4 ⊢ (𝜑 → {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)} = ∅)
17	16	adantr 482	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)} = ∅)
18	7, 17	eqtrd 2776	. 2 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (TermO‘𝐷) = ∅)
19	1, 2, 3, 18	initopropdlemlem 49741	1 ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∃!weu 2574  ∀wral 3055  {crab 3393  Vcvv 3433  ∅c0 4263  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  Hom chom 17226  Catccat 17625  Hom_f chomf 17627  comp_fccomf 17628  TermOctermo 17944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-1st 7933  df-2nd 7934  df-homf 17631  df-termo 17947

This theorem is referenced by:  zeroopropdlem  49744  termopropd  49746