Theorem fulltermc2 50014

Description: Given a full functor to a terminal category, the source category must not have empty hom-sets. (Contributed by Zhi Wang, 17-Oct-2025.) (Proof shortened by Zhi Wang, 6-Nov-2025.)

Hypotheses

Ref	Expression
fulltermc.b	⊢ 𝐵 = (Base‘𝐶)
fulltermc.h	⊢ 𝐻 = (Hom ‘𝐶)
fulltermc.d	⊢ (𝜑 → 𝐷 ∈ TermCat)
fulltermc2.f	⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)
fulltermc2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
fulltermc2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
fulltermc2	⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)

Proof of Theorem fulltermc2

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7366	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
2	1	eqeq1d 2743	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥𝐻𝑦) = ∅ ↔ (𝑋𝐻𝑦) = ∅))
3	2	notbid 320	. 2 ⊢ (𝑥 = 𝑋 → (¬ (𝑥𝐻𝑦) = ∅ ↔ ¬ (𝑋𝐻𝑦) = ∅))
4		oveq2 7367	. . . 4 ⊢ (𝑦 = 𝑌 → (𝑋𝐻𝑦) = (𝑋𝐻𝑌))
5	4	eqeq1d 2743	. . 3 ⊢ (𝑦 = 𝑌 → ((𝑋𝐻𝑦) = ∅ ↔ (𝑋𝐻𝑌) = ∅))
6	5	notbid 320	. 2 ⊢ (𝑦 = 𝑌 → (¬ (𝑋𝐻𝑦) = ∅ ↔ ¬ (𝑋𝐻𝑌) = ∅))
7		fulltermc2.f	. . 3 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)
8		fulltermc.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
9		fulltermc.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
10		fulltermc.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ TermCat)
11		fullfunc 17870	. . . . . 6 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
12	11	ssbri 5119	. . . . 5 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
13	7, 12	syl 17	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
14	8, 9, 10, 13	fulltermc 50013	. . 3 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ (𝑥𝐻𝑦) = ∅))
15	7, 14	mpbid 234	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ¬ (𝑥𝐻𝑦) = ∅)
16		fulltermc2.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
17		fulltermc2.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18	3, 6, 15, 16, 17	rspc2dv 3576	1 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∅c0 4263   class class class wbr 5074  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  Hom chom 17226   Func cfunc 17816   Full cful 17866  TermCatctermc 49974

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-rmo 3346  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-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-1st 7933  df-2nd 7934  df-map 8769  df-ixp 8840  df-cat 17629  df-cid 17630  df-func 17820  df-full 17868  df-thinc 49920  df-termc 49975

This theorem is referenced by: (None)