fullthinc - Mathbox for Zhi Wang

	Mathbox for Zhi Wang		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fullthinc		Structured version Visualization version GIF version

Theorem fullthinc 47753

Description: A functor to a thin category is full iff empty hom-sets are mapped 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)
fullthinc.f	⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)

**Assertion**
Ref	Expression
fullthinc	⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)))

Distinct variable groups: 𝑥,𝐵,𝑦 𝑥,𝐶,𝑦 𝑥,𝐷,𝑦 𝑥,𝐹,𝑦 𝑥,𝐺,𝑦 𝑥,𝐻,𝑦 𝑥,𝐽,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦)

Proof of Theorem fullthinc Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fullthinc.d	. 2 ⊢ (𝜑 → 𝐷 ∈ ThinCat)
2		fullthinc.f	. 2 ⊢ (𝜑 → 𝐹(𝐶 Func 𝐷)𝐺)
3		fullthinc.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
4		fullthinc.j	. . . . . 6 ⊢ 𝐽 = (Hom ‘𝐷)
5		fullthinc.h	. . . . . 6 ⊢ 𝐻 = (Hom ‘𝐶)
6	3, 4, 5	isfull2 17866	. . . . 5 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 ↔ (𝐹(𝐶 Func 𝐷)𝐺 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
7		foeq2 6801	. . . . . . . 8 ⊢ ((𝑥𝐻𝑦) = ∅ → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):∅–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
8		fo00 6868	. . . . . . . . 9 ⊢ ((𝑥𝐺𝑦):∅–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ ((𝑥𝐺𝑦) = ∅ ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))
9	8	simprbi 495	. . . . . . . 8 ⊢ ((𝑥𝐺𝑦):∅–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)
10	7, 9	syl6bi 252	. . . . . . 7 ⊢ ((𝑥𝐻𝑦) = ∅ → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))
11	10	com12 32	. . . . . 6 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))
12	11	2ralimi 3121	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))
13	6, 12	simplbiim 503	. . . 4 ⊢ (𝐹(𝐶 Full 𝐷)𝐺 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))
14	13	adantl 480	. . 3 ⊢ (((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ 𝐹(𝐶 Full 𝐷)𝐺) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))
15		simplr 765	. . . 4 ⊢ (((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) → 𝐹(𝐶 Func 𝐷)𝐺)
16		imor 849	. . . . . . . 8 ⊢ (((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) ↔ (¬ (𝑥𝐻𝑦) = ∅ ∨ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅))
17		simplr 765	. . . . . . . . . . . . . . . 16 ⊢ (((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐹(𝐶 Func 𝐷)𝐺)
18		simprl 767	. . . . . . . . . . . . . . . 16 ⊢ (((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
19		simprr 769	. . . . . . . . . . . . . . . 16 ⊢ (((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
20	3, 5, 4, 17, 18, 19	funcf2 17822	. . . . . . . . . . . . . . 15 ⊢ (((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
21	20	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
22		simpr 483	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → ¬ (𝑥𝐻𝑦) = ∅)
23	22	neqned 2945	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → (𝑥𝐻𝑦) ≠ ∅)
24		fdomne0 47603	. . . . . . . . . . . . . 14 ⊢ (((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝑥𝐻𝑦) ≠ ∅) → ((𝑥𝐺𝑦) ≠ ∅ ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ≠ ∅))
25	21, 23, 24	syl2anc 582	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → ((𝑥𝐺𝑦) ≠ ∅ ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ≠ ∅))
26	25	simprd 494	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ≠ ∅)
27		simplll 771	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → 𝐷 ∈ ThinCat)
28		eqid 2730	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝐷) = (Base‘𝐷)
29	17	adantr 479	. . . . . . . . . . . . . . 15 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → 𝐹(𝐶 Func 𝐷)𝐺)
30	3, 28, 29	funcf1 17820	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → 𝐹:𝐵⟶(Base‘𝐷))
31	18	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → 𝑥 ∈ 𝐵)
32	30, 31	ffvelcdmd 7086	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → (𝐹‘𝑥) ∈ (Base‘𝐷))
33	19	adantr 479	. . . . . . . . . . . . . 14 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → 𝑦 ∈ 𝐵)
34	30, 33	ffvelcdmd 7086	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → (𝐹‘𝑦) ∈ (Base‘𝐷))
35		eqidd 2731	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → (Base‘𝐷) = (Base‘𝐷))
36	4	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → 𝐽 = (Hom ‘𝐷))
37	27, 32, 34, 35, 36	thincn0eu 47739	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ≠ ∅ ↔ ∃!𝑓 𝑓 ∈ ((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
38	26, 37	mpbid 231	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → ∃!𝑓 𝑓 ∈ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
39		eusn 4733	. . . . . . . . . . 11 ⊢ (∃!𝑓 𝑓 ∈ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ ∃𝑓((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = {𝑓})
40	38, 39	sylib 217	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → ∃𝑓((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = {𝑓})
41	25	simpld 493	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → (𝑥𝐺𝑦) ≠ ∅)
42		foconst 6819	. . . . . . . . . . . . 13 ⊢ (((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶{𝑓} ∧ (𝑥𝐺𝑦) ≠ ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→{𝑓})
43		feq3 6699	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = {𝑓} → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶{𝑓}))
44	43	anbi1d 628	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = {𝑓} → (((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝑥𝐺𝑦) ≠ ∅) ↔ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶{𝑓} ∧ (𝑥𝐺𝑦) ≠ ∅)))
45		foeq3 6802	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = {𝑓} → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→{𝑓}))
46	44, 45	imbi12d 343	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = {𝑓} → ((((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝑥𝐺𝑦) ≠ ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))) ↔ (((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶{𝑓} ∧ (𝑥𝐺𝑦) ≠ ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→{𝑓})))
47	42, 46	mpbiri 257	. . . . . . . . . . . 12 ⊢ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = {𝑓} → (((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝑥𝐺𝑦) ≠ ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
48	47	exlimiv 1931	. . . . . . . . . . 11 ⊢ (∃𝑓((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = {𝑓} → (((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝑥𝐺𝑦) ≠ ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
49	48	imp 405	. . . . . . . . . 10 ⊢ ((∃𝑓((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = {𝑓} ∧ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ∧ (𝑥𝐺𝑦) ≠ ∅)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
50	40, 21, 41, 49	syl12anc 833	. . . . . . . . 9 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ¬ (𝑥𝐻𝑦) = ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
51	20	adantr 479	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
52		feq3 6699	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅ → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶∅))
53	52	adantl 480	. . . . . . . . . . . . 13 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶((𝐹‘𝑥)𝐽(𝐹‘𝑦)) ↔ (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶∅))
54	51, 53	mpbid 231	. . . . . . . . . . . 12 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶∅)
55		f00 6772	. . . . . . . . . . . 12 ⊢ ((𝑥𝐺𝑦):(𝑥𝐻𝑦)⟶∅ ↔ ((𝑥𝐺𝑦) = ∅ ∧ (𝑥𝐻𝑦) = ∅))
56	54, 55	sylib 217	. . . . . . . . . . 11 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → ((𝑥𝐺𝑦) = ∅ ∧ (𝑥𝐻𝑦) = ∅))
57	56	simprd 494	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → (𝑥𝐻𝑦) = ∅)
58	56	simpld 493	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → (𝑥𝐺𝑦) = ∅)
59		simpr 483	. . . . . . . . . 10 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)
60	8	biimpri 227	. . . . . . . . . . . 12 ⊢ (((𝑥𝐺𝑦) = ∅ ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → (𝑥𝐺𝑦):∅–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
61	60, 7	imbitrrid 245	. . . . . . . . . . 11 ⊢ ((𝑥𝐻𝑦) = ∅ → (((𝑥𝐺𝑦) = ∅ ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
62	61	imp 405	. . . . . . . . . 10 ⊢ (((𝑥𝐻𝑦) = ∅ ∧ ((𝑥𝐺𝑦) = ∅ ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
63	57, 58, 59, 62	syl12anc 833	. . . . . . . . 9 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
64	50, 63	jaodan 954	. . . . . . . 8 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (¬ (𝑥𝐻𝑦) = ∅ ∨ ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
65	16, 64	sylan2b 592	. . . . . . 7 ⊢ ((((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
66	65	ex 411	. . . . . 6 ⊢ (((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
67	66	ralimdvva 3202	. . . . 5 ⊢ ((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦))))
68	67	imp 405	. . . 4 ⊢ (((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐺𝑦):(𝑥𝐻𝑦)–onto→((𝐹‘𝑥)𝐽(𝐹‘𝑦)))
69	15, 68, 6	sylanbrc 581	. . 3 ⊢ (((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)) → 𝐹(𝐶 Full 𝐷)𝐺)
70	14, 69	impbida 797	. 2 ⊢ ((𝐷 ∈ ThinCat ∧ 𝐹(𝐶 Func 𝐷)𝐺) → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)))
71	1, 2, 70	syl2anc 582	1 ⊢ (𝜑 → (𝐹(𝐶 Full 𝐷)𝐺 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥𝐻𝑦) = ∅ → ((𝐹‘𝑥)𝐽(𝐹‘𝑦)) = ∅)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 843 = wceq 1539 ∃wex 1779 ∈ wcel 2104 ∃!weu 2560 ≠ wne 2938 ∀wral 3059 ∅c0 4321 {csn 4627 class class class wbr 5147 ⟶wf 6538 –onto→wfo 6540 ‘cfv 6542 (class class class)co 7411 Basecbs 17148 Hom chom 17212 Func cfunc 17808 Full cful 17857 ThinCatcthinc 47726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-map 8824 df-ixp 8894 df-func 17812 df-full 17859 df-thinc 47727

This theorem is referenced by: fullthinc2 47754 thincciso 47756

W3C validator