thincciso - Mathbox for Zhi Wang

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Theorem thincciso 47943

Description: Two thin categories are isomorphic iff the induced preorders are order-isomorphic. Example 3.26(2) of [Adamek] p. 33. (Contributed by Zhi Wang, 16-Oct-2024.)

**Hypotheses**
Ref	Expression
thincciso.c	⊢ 𝐶 = (CatCat‘𝑈)
thincciso.b	⊢ 𝐵 = (Base‘𝐶)
thincciso.r	⊢ 𝑅 = (Base‘𝑋)
thincciso.s	⊢ 𝑆 = (Base‘𝑌)
thincciso.h	⊢ 𝐻 = (Hom ‘𝑋)
thincciso.j	⊢ 𝐽 = (Hom ‘𝑌)
thincciso.u	⊢ (𝜑 → 𝑈 ∈ 𝑉)
thincciso.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
thincciso.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
thincciso.xt	⊢ (𝜑 → 𝑋 ∈ ThinCat)
thincciso.yt	⊢ (𝜑 → 𝑌 ∈ ThinCat)

**Assertion**
Ref	Expression
thincciso	⊢ (𝜑 → (𝑋( ≃_𝑐 ‘𝐶)𝑌 ↔ ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)))

Distinct variable groups: 𝐶,𝑓,𝑥,𝑦 𝑓,𝐻,𝑥,𝑦 𝑓,𝐽,𝑥,𝑦 𝑅,𝑓,𝑥,𝑦 𝑆,𝑓 𝑓,𝑋,𝑥,𝑦 𝑓,𝑌,𝑥,𝑦 𝜑,𝑓,𝑥,𝑦 Allowed substitution hints: 𝐵(𝑥,𝑦,𝑓) 𝑆(𝑥,𝑦) 𝑈(𝑥,𝑦,𝑓) 𝑉(𝑥,𝑦,𝑓)

Proof of Theorem thincciso Dummy variables 𝑎 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2726	. . 3 ⊢ (Iso‘𝐶) = (Iso‘𝐶)
2		thincciso.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3		thincciso.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
4		thincciso.c	. . . . 5 ⊢ 𝐶 = (CatCat‘𝑈)
5	4	catccat 18070	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 17	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
7		thincciso.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		thincciso.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	1, 2, 6, 7, 8	cic 17755	. 2 ⊢ (𝜑 → (𝑋( ≃_𝑐 ‘𝐶)𝑌 ↔ ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
10		opex 5457	. . . . . . 7 ⊢ ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ V
11	10	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ V)
12		biimp 214	. . . . . . . . . . . . 13 ⊢ (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → ((𝑥𝐻𝑦) = ∅ → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅))
13	12	2ralimi 3117	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅))
14	13	ad2antrl 725	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅))
15		thincciso.r	. . . . . . . . . . . 12 ⊢ 𝑅 = (Base‘𝑋)
16		thincciso.j	. . . . . . . . . . . 12 ⊢ 𝐽 = (Hom ‘𝑌)
17		thincciso.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (Hom ‘𝑋)
18		thincciso.yt	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑌 ∈ ThinCat)
19	18	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑌 ∈ ThinCat)
20		eqid 2726	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))) = (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))
21		thincciso.s	. . . . . . . . . . . . . 14 ⊢ 𝑆 = (Base‘𝑌)
22		thincciso.xt	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ∈ ThinCat)
23	22	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑋 ∈ ThinCat)
24	23	thinccd 47919	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑋 ∈ Cat)
25		simprr 770	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑓:𝑅–1-1-onto→𝑆)
26		f1of 6827	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑅–1-1-onto→𝑆 → 𝑓:𝑅⟶𝑆)
27	25, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑓:𝑅⟶𝑆)
28		biimpr 219	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
29	28	2ralimi 3117	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
30	29	ad2antrl 725	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
31	15, 21, 17, 16, 24, 19, 27, 20, 30	functhinc 47939	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → (𝑓(𝑋 Func 𝑌)(𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))) ↔ (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))) = (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))))
32	20, 31	mpbiri 258	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑓(𝑋 Func 𝑌)(𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))))
33	15, 16, 17, 19, 32	fullthinc 47940	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → (𝑓(𝑋 Full 𝑌)(𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))) ↔ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅)))
34	14, 33	mpbird 257	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑓(𝑋 Full 𝑌)(𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))))
35		df-br 5142	. . . . . . . . . 10 ⊢ (𝑓(𝑋 Full 𝑌)(𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))) ↔ ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ (𝑋 Full 𝑌))
36	34, 35	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ (𝑋 Full 𝑌))
37	23, 32	thincfth 47942	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑓(𝑋 Faith 𝑌)(𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))))
38		df-br 5142	. . . . . . . . . 10 ⊢ (𝑓(𝑋 Faith 𝑌)(𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))) ↔ ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ (𝑋 Faith 𝑌))
39	37, 38	sylib 217	. . . . . . . . 9 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ (𝑋 Faith 𝑌))
40	36, 39	elind 4189	. . . . . . . 8 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
41		vex 3472	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
42	15	fvexi 6899	. . . . . . . . . . . 12 ⊢ 𝑅 ∈ V
43	42, 42	mpoex 8065	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))) ∈ V
44	41, 43	op1st 7982	. . . . . . . . . 10 ⊢ (1^st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩) = 𝑓
45		f1oeq1 6815	. . . . . . . . . 10 ⊢ ((1^st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩) = 𝑓 → ((1^st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩):𝑅–1-1-onto→𝑆 ↔ 𝑓:𝑅–1-1-onto→𝑆))
46	44, 45	ax-mp 5	. . . . . . . . 9 ⊢ ((1^st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩):𝑅–1-1-onto→𝑆 ↔ 𝑓:𝑅–1-1-onto→𝑆)
47	25, 46	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → (1^st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩):𝑅–1-1-onto→𝑆)
48	40, 47	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → (⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩):𝑅–1-1-onto→𝑆))
49	4, 2, 15, 21, 3, 7, 8, 1	catciso 18073	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ (𝑋(Iso‘𝐶)𝑌) ↔ (⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩):𝑅–1-1-onto→𝑆)))
50	49	biimpar 477	. . . . . . 7 ⊢ ((𝜑 ∧ (⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩):𝑅–1-1-onto→𝑆)) → ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ (𝑋(Iso‘𝐶)𝑌))
51	48, 50	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ (𝑋(Iso‘𝐶)𝑌))
52		eleq1 2815	. . . . . 6 ⊢ (𝑎 = ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ → (𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ ⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ (𝑋(Iso‘𝐶)𝑌)))
53	11, 51, 52	spcedv 3582	. . . . 5 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌))
54	53	ex 412	. . . 4 ⊢ (𝜑 → ((∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆) → ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
55	54	exlimdv 1928	. . 3 ⊢ (𝜑 → (∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆) → ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
56		fvexd 6900	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1^st ‘𝑎) ∈ V)
57		relfull 17870	. . . . . . . . . 10 ⊢ Rel (𝑋 Full 𝑌)
58	4, 2, 15, 21, 3, 7, 8, 1	catciso 18073	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ (𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝑎):𝑅–1-1-onto→𝑆)))
59	58	biimpa 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1^st ‘𝑎):𝑅–1-1-onto→𝑆))
60	59	simpld 494	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
61	60	elin1d 4193	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑎 ∈ (𝑋 Full 𝑌))
62		1st2ndbr 8027	. . . . . . . . . 10 ⊢ ((Rel (𝑋 Full 𝑌) ∧ 𝑎 ∈ (𝑋 Full 𝑌)) → (1^st ‘𝑎)(𝑋 Full 𝑌)(2^nd ‘𝑎))
63	57, 61, 62	sylancr 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1^st ‘𝑎)(𝑋 Full 𝑌)(2^nd ‘𝑎))
64	18	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑌 ∈ ThinCat)
65		fullfunc 17868	. . . . . . . . . . . 12 ⊢ (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
66	65	ssbri 5186	. . . . . . . . . . 11 ⊢ ((1^st ‘𝑎)(𝑋 Full 𝑌)(2^nd ‘𝑎) → (1^st ‘𝑎)(𝑋 Func 𝑌)(2^nd ‘𝑎))
67	63, 66	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1^st ‘𝑎)(𝑋 Func 𝑌)(2^nd ‘𝑎))
68	15, 16, 17, 64, 67	fullthinc 47940	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ((1^st ‘𝑎)(𝑋 Full 𝑌)(2^nd ‘𝑎) ↔ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅)))
69	63, 68	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅))
70	67	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1^st ‘𝑎)(𝑋 Func 𝑌)(2^nd ‘𝑎))
71		simprl 768	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
72		simprr 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
73	15, 17, 16, 70, 71, 72	funcf2 17827	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2^nd ‘𝑎)𝑦):(𝑥𝐻𝑦)⟶(((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)))
74	73	f002 47794	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
75	74	ralrimivva 3194	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
76		2ralbiim 3126	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅) ↔ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅)))
77	69, 75, 76	sylanbrc 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅))
78	59	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1^st ‘𝑎):𝑅–1-1-onto→𝑆)
79	77, 78	jca 511	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅) ∧ (1^st ‘𝑎):𝑅–1-1-onto→𝑆))
80		fveq1 6884	. . . . . . . . . . 11 ⊢ (𝑓 = (1^st ‘𝑎) → (𝑓‘𝑥) = ((1^st ‘𝑎)‘𝑥))
81		fveq1 6884	. . . . . . . . . . 11 ⊢ (𝑓 = (1^st ‘𝑎) → (𝑓‘𝑦) = ((1^st ‘𝑎)‘𝑦))
82	80, 81	oveq12d 7423	. . . . . . . . . 10 ⊢ (𝑓 = (1^st ‘𝑎) → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)))
83	82	eqeq1d 2728	. . . . . . . . 9 ⊢ (𝑓 = (1^st ‘𝑎) → (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ ↔ (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅))
84	83	bibi2d 342	. . . . . . . 8 ⊢ (𝑓 = (1^st ‘𝑎) → (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ↔ ((𝑥𝐻𝑦) = ∅ ↔ (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅)))
85	84	2ralbidv 3212	. . . . . . 7 ⊢ (𝑓 = (1^st ‘𝑎) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ↔ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅)))
86		f1oeq1 6815	. . . . . . 7 ⊢ (𝑓 = (1^st ‘𝑎) → (𝑓:𝑅–1-1-onto→𝑆 ↔ (1^st ‘𝑎):𝑅–1-1-onto→𝑆))
87	85, 86	anbi12d 630	. . . . . 6 ⊢ (𝑓 = (1^st ‘𝑎) → ((∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆) ↔ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1^st ‘𝑎)‘𝑥)𝐽((1^st ‘𝑎)‘𝑦)) = ∅) ∧ (1^st ‘𝑎):𝑅–1-1-onto→𝑆)))
88	56, 79, 87	spcedv 3582	. . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆))
89	88	ex 412	. . . 4 ⊢ (𝜑 → (𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) → ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)))
90	89	exlimdv 1928	. . 3 ⊢ (𝜑 → (∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) → ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)))
91	55, 90	impbid 211	. 2 ⊢ (𝜑 → (∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆) ↔ ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
92	9, 91	bitr4d 282	1 ⊢ (𝜑 → (𝑋( ≃_𝑐 ‘𝐶)𝑌 ↔ ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ∩ cin 3942 ∅c0 4317 ⟨cop 4629 class class class wbr 5141 × cxp 5667 Rel wrel 5674 ⟶wf 6533 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 1^st c1st 7972 2^nd c2nd 7973 Basecbs 17153 Hom chom 17217 Catccat 17617 Isociso 17702 ≃_𝑐 ccic 17751 Func cfunc 17813 Full cful 17864 Faith cfth 17865 CatCatccatc 18060 ThinCatcthinc 47913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-fz 13491 df-struct 17089 df-slot 17124 df-ndx 17136 df-base 17154 df-hom 17230 df-cco 17231 df-cat 17621 df-cid 17622 df-sect 17703 df-inv 17704 df-iso 17705 df-cic 17752 df-func 17817 df-idfu 17818 df-cofu 17819 df-full 17866 df-fth 17867 df-catc 18061 df-thinc 47914

This theorem is referenced by: (None)

W3C validator