Theorem termopropdlem 49600

Description: Lemma for termopropd 49603. (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 17924	. 2 ⊢ TermO Fn Cat
2		initopropdlem.1	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ V)
3		ssv 3960	. 2 ⊢ Cat ⊆ V
4		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → 𝐷 ∈ Cat)
5		eqid 2737	. . . 4 ⊢ (Base‘𝐷) = (Base‘𝐷)
6		eqid 2737	. . . 4 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
7	4, 5, 6	termoval 17930	. . 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 6834	. . . . . . . . 9 ⊢ (¬ 𝐶 ∈ V → (Homf ‘𝐶) = ∅)
10	2, 9	syl 17	. . . . . . . 8 ⊢ (𝜑 → (Homf ‘𝐶) = ∅)
11	8, 10	eqtr3d 2774	. . . . . . 7 ⊢ (𝜑 → (Homf ‘𝐷) = ∅)
12		homf0 49368	. . . . . . 7 ⊢ ((Base‘𝐷) = ∅ ↔ (Homf ‘𝐷) = ∅)
13	11, 12	sylibr 234	. . . . . 6 ⊢ (𝜑 → (Base‘𝐷) = ∅)
14	13	rabeqdv 3416	. . . . 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 4340	. . . . 5 ⊢ {𝑎 ∈ ∅ ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)} = ∅
16	14, 15	eqtrdi 2788	. . . 4 ⊢ (𝜑 → {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)} = ∅)
17	16	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → {𝑎 ∈ (Base‘𝐷) ∣ ∀𝑏 ∈ (Base‘𝐷)∃!ℎ ℎ ∈ (𝑏(Hom ‘𝐷)𝑎)} = ∅)
18	7, 17	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝐷 ∈ Cat) → (TermO‘𝐷) = ∅)
19	1, 2, 3, 18	initopropdlemlem 49598	1 ⊢ (𝜑 → (TermO‘𝐶) = (TermO‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃!weu 2569  ∀wral 3052  {crab 3401  Vcvv 3442  ∅c0 4287  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  Hom chom 17200  Catccat 17599  Hom_f chomf 17601  comp_fccomf 17602  TermOctermo 17918

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-homf 17605  df-termo 17921

This theorem is referenced by:  zeroopropdlem  49601  termopropd  49603