Theorem thincciso 49438

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 17126. (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 2729	. . 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 18015	. . . 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 17706	. 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
10		opex 5407	. . . . . . 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 215	. . . . . . . . . . . . 13 ⊢ (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → ((𝑥𝐻𝑦) = ∅ → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅))
13	12	2ralimi 3099	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅))
14	13	ad2antrl 728	. . . . . . . . . . 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 2729	. . . . . . . . . . . . 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 49408	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ∧ 𝑓:𝑅–1-1-onto→𝑆)) → 𝑋 ∈ Cat)
25		simprr 772	. . . . . . . . . . . . . . 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 6764	. . . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . 16 ⊢ (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
29	28	2ralimi 3099	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
30	29	ad2antrl 728	. . . . . . . . . . . . . 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 49433	. . . . . . . . . . . . 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 49435	. . . . . . . . . . 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 5093	. . . . . . . . . 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 218	. . . . . . . . 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 49437	. . . . . . . . . 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 5093	. . . . . . . . . 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 218	. . . . . . . . 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 4151	. . . . . . . 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 3440	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
42	15	fvexi 6836	. . . . . . . . . . . 12 ⊢ 𝑅 ∈ V
43	42, 42	mpoex 8014	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))) ∈ V
44	41, 43	op1st 7932	. . . . . . . . . 10 ⊢ (1st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩) = 𝑓
45		f1oeq1 6752	. . . . . . . . . 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 234	. . . . . . . 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 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)) ∧ (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 18018	. . . . . . . 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 477	. . . . . . 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 591	. . . . . 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 2816	. . . . . 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 3553	. . . . 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 1933	. . 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 6837	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st ‘𝑎) ∈ V)
57		relfull 17817	. . . . . . . . . 10 ⊢ Rel (𝑋 Full 𝑌)
58	4, 2, 15, 21, 3, 7, 8, 1	catciso 18018	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ (𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘𝑎):𝑅–1-1-onto→𝑆)))
59	58	biimpa 476	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘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 4155	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑎 ∈ (𝑋 Full 𝑌))
62		1st2ndbr 7977	. . . . . . . . . 10 ⊢ ((Rel (𝑋 Full 𝑌) ∧ 𝑎 ∈ (𝑋 Full 𝑌)) → (1st ‘𝑎)(𝑋 Full 𝑌)(2nd ‘𝑎))
63	57, 61, 62	sylancr 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st ‘𝑎)(𝑋 Full 𝑌)(2nd ‘𝑎))
64	18	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑌 ∈ ThinCat)
65		fullfunc 17815	. . . . . . . . . . . 12 ⊢ (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
66	65	ssbri 5137	. . . . . . . . . . 11 ⊢ ((1st ‘𝑎)(𝑋 Full 𝑌)(2nd ‘𝑎) → (1st ‘𝑎)(𝑋 Func 𝑌)(2nd ‘𝑎))
67	63, 66	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st ‘𝑎)(𝑋 Func 𝑌)(2nd ‘𝑎))
68	15, 16, 17, 64, 67	fullthinc 49435	. . . . . . . . 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 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ → (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅))
70	67	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → (1st ‘𝑎)(𝑋 Func 𝑌)(2nd ‘𝑎))
71		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑥 ∈ 𝑅)
72		simprr 772	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → 𝑦 ∈ 𝑅)
73	15, 17, 16, 70, 71, 72	funcf2 17775	. . . . . . . . . 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 48838	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
75	74	ralrimivva 3172	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
76		2ralbiim 3108	. . . . . . . 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 583	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅))
78	59	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st ‘𝑎):𝑅–1-1-onto→𝑆)
79	77, 78	jca 511	. . . . . 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 6821	. . . . . . . . . . 11 ⊢ (𝑓 = (1st ‘𝑎) → (𝑓‘𝑥) = ((1st ‘𝑎)‘𝑥))
81		fveq1 6821	. . . . . . . . . . 11 ⊢ (𝑓 = (1st ‘𝑎) → (𝑓‘𝑦) = ((1st ‘𝑎)‘𝑦))
82	80, 81	oveq12d 7367	. . . . . . . . . 10 ⊢ (𝑓 = (1st ‘𝑎) → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)))
83	82	eqeq1d 2731	. . . . . . . . 9 ⊢ (𝑓 = (1st ‘𝑎) → (((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅ ↔ (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅))
84	83	bibi2d 342	. . . . . . . 8 ⊢ (𝑓 = (1st ‘𝑎) → (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = ∅) ↔ ((𝑥𝐻𝑦) = ∅ ↔ (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅)))
85	84	2ralbidv 3193	. . . . . . 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 6752	. . . . . . 7 ⊢ (𝑓 = (1st ‘𝑎) → (𝑓:𝑅–1-1-onto→𝑆 ↔ (1st ‘𝑎):𝑅–1-1-onto→𝑆))
87	85, 86	anbi12d 632	. . . . . 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 3553	. . . . 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 1933	. . 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 212	. 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)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∀wral 3044  Vcvv 3436   ∩ cin 3902  ∅c0 4284  ⟨cop 4583   class class class wbr 5092   × cxp 5617  Rel wrel 5624  ⟶wf 6478  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  1^st c1st 7922  2^nd c2nd 7923  Basecbs 17120  Hom chom 17172  Catccat 17570  Isociso 17653   ≃_𝑐 ccic 17702   Func cfunc 17761   Full cful 17811   Faith cfth 17812  CatCatccatc 18005  ThinCatcthinc 49402

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-supp 8094  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-er 8625  df-map 8755  df-ixp 8825  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-3 12192  df-4 12193  df-5 12194  df-6 12195  df-7 12196  df-8 12197  df-9 12198  df-n0 12385  df-z 12472  df-dec 12592  df-uz 12736  df-fz 13411  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-hom 17185  df-cco 17186  df-cat 17574  df-cid 17575  df-sect 17654  df-inv 17655  df-iso 17656  df-cic 17703  df-func 17765  df-idfu 17766  df-cofu 17767  df-full 17813  df-fth 17814  df-catc 18006  df-thinc 49403

This theorem is referenced by:  thinccisod  49439