Theorem thincciso 49127

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 17254. (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 2736	. . 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 18154	. . . 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 17844	. 2 ⊢ (𝜑 → (𝑋( ≃𝑐 ‘𝐶)𝑌 ↔ ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
10		opex 5468	. . . . . . 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 3122	. . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 49097	. . . . . . . . . . . . . 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 6847	. . . . . . . . . . . . . . 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 3122	. . . . . . . . . . . . . . 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 49122	. . . . . . . . . . . . 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 49124	. . . . . . . . . . 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 5143	. . . . . . . . . 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 49126	. . . . . . . . . 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 5143	. . . . . . . . . 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 4199	. . . . . . . 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 3483	. . . . . . . . . . 11 ⊢ 𝑓 ∈ V
42	15	fvexi 6919	. . . . . . . . . . . 12 ⊢ 𝑅 ∈ V
43	42, 42	mpoex 8105	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤)))) ∈ V
44	41, 43	op1st 8023	. . . . . . . . . 10 ⊢ (1st ‘⟨𝑓, (𝑧 ∈ 𝑅, 𝑤 ∈ 𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓‘𝑧)𝐽(𝑓‘𝑤))))⟩) = 𝑓
45		f1oeq1 6835	. . . . . . . . . 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 18157	. . . . . . . 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 2828	. . . . . 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 3597	. . . . 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 1932	. . 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 6920	. . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st ‘𝑎) ∈ V)
57		relfull 17956	. . . . . . . . . 10 ⊢ Rel (𝑋 Full 𝑌)
58	4, 2, 15, 21, 3, 7, 8, 1	catciso 18157	. . . . . . . . . . . . 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 4203	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑎 ∈ (𝑋 Full 𝑌))
62		1st2ndbr 8068	. . . . . . . . . 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 17954	. . . . . . . . . . . 12 ⊢ (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
66	65	ssbri 5187	. . . . . . . . . . 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 49124	. . . . . . . . 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 17914	. . . . . . . . . 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 48768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) → ((((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
75	74	ralrimivva 3201	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑅 ((((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
76		2ralbiim 3131	. . . . . . . 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 6904	. . . . . . . . . . 11 ⊢ (𝑓 = (1st ‘𝑎) → (𝑓‘𝑥) = ((1st ‘𝑎)‘𝑥))
81		fveq1 6904	. . . . . . . . . . 11 ⊢ (𝑓 = (1st ‘𝑎) → (𝑓‘𝑦) = ((1st ‘𝑎)‘𝑦))
82	80, 81	oveq12d 7450	. . . . . . . . . 10 ⊢ (𝑓 = (1st ‘𝑎) → ((𝑓‘𝑥)𝐽(𝑓‘𝑦)) = (((1st ‘𝑎)‘𝑥)𝐽((1st ‘𝑎)‘𝑦)))
83	82	eqeq1d 2738	. . . . . . . . 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 3220	. . . . . . 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 6835	. . . . . . 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 3597	. . . . 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 1932	. . 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 1539  ∃wex 1778   ∈ wcel 2107  ∀wral 3060  Vcvv 3479   ∩ cin 3949  ∅c0 4332  ⟨cop 4631   class class class wbr 5142   × cxp 5682  Rel wrel 5689  ⟶wf 6556  –1-1-onto→wf1o 6559  ‘cfv 6560  (class class class)co 7432   ∈ cmpo 7434  1^st c1st 8013  2^nd c2nd 8014  Basecbs 17248  Hom chom 17309  Catccat 17708  Isociso 17791   ≃_𝑐 ccic 17840   Func cfunc 17900   Full cful 17950   Faith cfth 17951  CatCatccatc 18144  ThinCatcthinc 49091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-fz 13549  df-struct 17185  df-slot 17220  df-ndx 17232  df-base 17249  df-hom 17322  df-cco 17323  df-cat 17712  df-cid 17713  df-sect 17792  df-inv 17793  df-iso 17794  df-cic 17841  df-func 17904  df-idfu 17905  df-cofu 17906  df-full 17952  df-fth 17953  df-catc 18145  df-thinc 49092

This theorem is referenced by:  thinccisod  49128