Theorem thincciso 50696

Description: Two thin categories are isomorphic iff the induced preorders are order-isomorphic. Example 3.26(2) of [Adamek] p. 33. Note that "thincciso.u" is redundant thanks to elbasfv 17402. (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 2760	. . 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 18292	. . . 4 ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
6	3, 5	syl 18	. . 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 17983	. 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
10		opex 5432	. . . . . . 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 218	. . . . . . . . . . . . 13 ⊢ (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → ((𝑥𝐻𝑦) = ∅ → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅))
13	12	2ralimi 3132	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅))
14	13	ad2antrl 741	. . . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑌 ∈ ThinCat)
20		eqid 2760	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑋 ∈ ThinCat)
24	23	thinccatd 50666	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑋 ∈ Cat)
25		simprr 785	. . . . . . . . . . . . . . 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 6829	. . . . . . . . . . . . . . 15 ⊢ (𝑓:𝑅–1-1-onto→𝑆 → 𝑓:𝑅⟶𝑆)
27	25, 26	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑓:𝑅⟶𝑆)
28		biimpr 223	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
29	28	2ralimi 3132	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
30	29	ad2antrl 741	. . . . . . . . . . . . . 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 50691	. . . . . . . . . . . . 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 261	. . . . . . . . . . . 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 50693	. . . . . . . . . . 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 260	. . . . . . . . . 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 5103	. . . . . . . . . 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 221	. . . . . . . . 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 50695	. . . . . . . . . 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 5103	. . . . . . . . . 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 221	. . . . . . . . 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 4096	. . . . . . . 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 3454	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
42	15	fvexi 6904	. . . . . . . . . . . 12 ⊢ 𝑅 ∈ V
43	42, 42	mpoex 8096	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))) ∈ V
44	41, 43	op1st 8013	. . . . . . . . . 10 ⊢ (1st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩) = 𝑓
45		f1oeq1 6817	. . . . . . . . . 10 ⊢ ((1st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩) = 𝑓 → ((1st ‘⟨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 ⊢ ((1st ‘⟨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 237	. . . . . . . 8 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → (1st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩):𝑅–1-1-onto→𝑆)
48	40, 47	jca 521	. . . . . . 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)) ∧ (1st ‘⟨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 18295	. . . . . . . 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)) ∧ (1st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩):𝑅–1-1-onto→𝑆)))
50	49	biimpar 483	. . . . . . 7 ⊢ ((𝜑 ∧ (⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘⟨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 603	. . . . . 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 2848	. . . . . 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 3552	. . . . 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 418	. . . 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 1965	. . 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 6905	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st ‘𝑎) ∈ V)
57		relfull 18094	. . . . . . . . . 10 ⊢ Rel (𝑋 Full 𝑌)
58	4, 2, 15, 21, 3, 7, 8, 1	catciso 18295	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ (𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝑎):𝑅–1-1-onto→𝑆)))
59	58	biimpa 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝑎):𝑅–1-1-onto→𝑆))
60	59	simpld 500	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
61	60	elin1d 4100	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑎 ∈ (𝑋 Full 𝑌))
62		1st2ndbr 8057	. . . . . . . . . 10 ⊢ ((Rel (𝑋 Full 𝑌) ∧ 𝑎 ∈ (𝑋 Full 𝑌)) → (1st ‘𝑎)(𝑋 Full 𝑌)(2nd ‘𝑎))
63	57, 61, 62	sylancr 599	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st ‘𝑎)(𝑋 Full 𝑌)(2nd ‘𝑎))
64	18	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑌 ∈ ThinCat)
65		fullfunc 18092	. . . . . . . . . . . 12 ⊢ (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
66	65	ssbri 5149	. . . . . . . . . . 11 ⊢ ((1st ‘𝑎)(𝑋 Full 𝑌)(2nd ‘𝑎) → (1st ‘𝑎)(𝑋 Func 𝑌)(2nd ‘𝑎))
67	63, 66	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st ‘𝑎)(𝑋 Func 𝑌)(2nd ‘𝑎))
68	15, 16, 17, 64, 67	fullthinc 50693	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ((1st ‘𝑎)(𝑋 Full 𝑌)(2nd ‘𝑎) ↔ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅)))
69	63, 68	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅))
70	67	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1st ‘𝑎)(𝑋 Func 𝑌)(2nd ‘𝑎))
71		simprl 783	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
72		simprr 785	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
73	15, 17, 16, 70, 71, 72	funcf2 18052	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (𝑥(2nd ‘𝑎)𝑦):(𝑥𝐻𝑦)⟶(((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)))
74	73	f002 50073	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
75	74	ralrimivva 3205	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
76		2ralbiim 3141	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅) ↔ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅) ∧ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅)))
77	69, 75, 76	sylanbrc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅))
78	59	simprd 501	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st ‘𝑎):𝑅–1-1-onto→𝑆)
79	77, 78	jca 521	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅) ∧ (1st ‘𝑎):𝑅–1-1-onto→𝑆))
80		fveq1 6889	. . . . . . . . . . 11 ⊢ (𝑓 = (1st ‘𝑎) → (𝑓‘𝑥) = ((1st ‘𝑎)‘𝑥))
81		fveq1 6889	. . . . . . . . . . 11 ⊢ (𝑓 = (1st ‘𝑎) → (𝑓‘𝑦) = ((1st ‘𝑎)‘𝑦))
82	80, 81	oveq12d 7442	. . . . . . . . . 10 ⊢ (𝑓 = (1st ‘𝑎) → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)))
83	82	eqeq1d 2762	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑎) → (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ ↔ (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅))
84	83	bibi2d 345	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑎) → (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ↔ ((𝑥𝐻𝑦) = ∅ ↔ (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅)))
85	84	2ralbidv 3226	. . . . . . 7 ⊢ (𝑓 = (1st ‘𝑎) → (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ↔ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅)))
86		f1oeq1 6817	. . . . . . 7 ⊢ (𝑓 = (1st ‘𝑎) → (𝑓:𝑅–1-1-onto→𝑆 ↔ (1st ‘𝑎):𝑅–1-1-onto→𝑆))
87	85, 86	anbi12d 644	. . . . . 6 ⊢ (𝑓 = (1st ‘𝑎) → ((∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆) ↔ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅) ∧ (1st ‘𝑎):𝑅–1-1-onto→𝑆)))
88	56, 79, 87	spcedv 3552	. . . . 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 418	. . . 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 1965	. . 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 215	. 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 285	1 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑓(∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ∩ cin 4088  ∅c0 4277  ⟨cop 4834   class class class wbr 5102   × cxp 5649  Rel wrel 5656  ⟶wf 6539  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7424   ∈ cmpo 7426  1^st c1st 8003  2^nd c2nd 8004  Basecbs 17396  Hom chom 17448  Catccat 17847  Isociso 17930   ≃_𝑐 ccic 17979   Func cfunc 18038   Full cful 18088   Faith cfth 18089  CatCatccatc 18282  ThinCatcthinc 50660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755  ax-cnex 11263  ax-resscn 11264  ax-1cn 11265  ax-icn 11266  ax-addcl 11267  ax-addrcl 11268  ax-mulcl 11269  ax-mulrcl 11270  ax-mulcom 11271  ax-addass 11272  ax-mulass 11273  ax-distr 11274  ax-i2m1 11275  ax-1ne0 11276  ax-1rid 11277  ax-rnegex 11278  ax-rrecex 11279  ax-cnre 11280  ax-pre-lttri 11281  ax-pre-lttrn 11282  ax-pre-ltadd 11283  ax-pre-mulgt0 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7882  df-1st 8005  df-2nd 8006  df-supp 8184  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-rdg 8425  df-1o 8483  df-er 8724  df-map 8856  df-ixp 8933  df-en 8981  df-dom 8982  df-sdom 8983  df-fin 8984  df-pnf 11352  df-mnf 11353  df-xr 11354  df-ltxr 11355  df-le 11356  df-sub 11550  df-neg 11551  df-nn 12343  df-2 12412  df-3 12413  df-4 12414  df-5 12415  df-6 12416  df-7 12417  df-8 12418  df-9 12419  df-n0 12614  df-z 12701  df-dec 12822  df-uz 12973  df-fz 13647  df-struct 17334  df-slot 17369  df-ndx 17381  df-base 17397  df-hom 17461  df-cco 17462  df-cat 17851  df-cid 17852  df-sect 17931  df-inv 17932  df-iso 17933  df-cic 17980  df-func 18042  df-idfu 18043  df-cofu 18044  df-full 18090  df-fth 18091  df-catc 18283  df-thinc 50661

This theorem is used by:  thinccisod  50697