Theorem fullthinc2 46688

Description: A full functor to a thin category maps empty hom-sets to empty hom-sets. (Contributed by Zhi Wang, 1-Oct-2024.)

Hypotheses

Ref	Expression
fullthinc.b	⊢ 𝐵 = (Base‘𝐶)
fullthinc.j	⊢ 𝐽 = (Hom ‘𝐷)
fullthinc.h	⊢ 𝐻 = (Hom ‘𝐶)
fullthinc.d	⊢ (𝜑 → 𝐷 ∈ ThinCat)
fullthinc2.f	⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)
fullthinc2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
fullthinc2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
fullthinc2	⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))

Proof of Theorem fullthinc2

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

Step	Hyp	Ref	Expression
1		fullthinc2.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
2		fullthinc2.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
3		fullthinc2.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐶 Full 𝐷)𝐺)
4		fullthinc.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5		fullthinc.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝐷)
6		fullthinc.h	. . . . 5 ⊢ 𝐻 = (Hom ‘𝐶)
7		fullthinc.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
8		fullfunc 17719	. . . . . . 7 ⊢ (𝐶 Full 𝐷) ⊆ (𝐶 Func 𝐷)
9	8	ssbri 5137	. . . . . 6 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → 𝐹(𝐶 Func 𝐷)𝐺)
10	3, 9	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
11	4, 5, 6, 7, 10	fullthinc 46687	. . . 4 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)))
12	3, 11	mpbid 231	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))
13		oveq12 7346	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝑥𝐻𝑦) = (𝑋𝐻𝑌))
14	13	eqeq1d 2738	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝑥𝐻𝑦) = ∅ ↔ (𝑋𝐻𝑌) = ∅))
15		simpl 483	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑥 = 𝑋)
16	15	fveq2d 6829	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹‘𝑥) = (𝐹‘𝑋))
17		simpr 485	. . . . . . . . 9 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → 𝑦 = 𝑌)
18	17	fveq2d 6829	. . . . . . . 8 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (𝐹‘𝑦) = (𝐹‘𝑌))
19	16, 18	oveq12d 7355	. . . . . . 7 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
20	19	eqeq1d 2738	. . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
21	14, 20	imbi12d 344	. . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑦 = 𝑌) → (((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) ↔ ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅)))
22	21	rspc2gv 3578	. . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅)))
23	22	imp 407	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) → ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
24	1, 2, 12, 23	syl21anc 835	. 2 ⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ → ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))
25	4, 6, 5, 10, 1, 2	funcf2 17680	. . 3 ⊢ (𝜑 → (𝑋𝐺𝑌):(𝑋𝐻𝑌)⟶((𝐹‘𝑋)𝐽(𝐹‘𝑌)))
26	25	f002 46541	. 2 ⊢ (𝜑 → (((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅ → (𝑋𝐻𝑌) = ∅))
27	24, 26	impbid 211	1 ⊢ (𝜑 → ((𝑋𝐻𝑌) = ∅ ↔ ((𝐹‘𝑋)𝐽(𝐹‘𝑌)) = ∅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3061  ∅c0 4269   class class class wbr 5092  ‘cfv 6479  (class class class)co 7337  Basecbs 17009  Hom chom 17070   Func cfunc 17666   Full cful 17715  ThinCatcthinc 46660

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  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 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-map 8688  df-ixp 8757  df-func 17670  df-full 17717  df-thinc 46661

This theorem is referenced by: (None)