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Theorem thincciso 48133
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 (𝜑𝑈𝑉)
thincciso.x (𝜑𝑋𝐵)
thincciso.y (𝜑𝑌𝐵)
thincciso.xt (𝜑𝑋 ∈ ThinCat)
thincciso.yt (𝜑𝑌 ∈ ThinCat)
Assertion
Ref Expression
thincciso (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ ∃𝑓(∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)))
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.
StepHypRef Expression
1 eqid 2728 . . 3 (Iso‘𝐶) = (Iso‘𝐶)
2 thincciso.b . . 3 𝐵 = (Base‘𝐶)
3 thincciso.u . . . 4 (𝜑𝑈𝑉)
4 thincciso.c . . . . 5 𝐶 = (CatCat‘𝑈)
54catccat 18104 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
63, 5syl 17 . . 3 (𝜑𝐶 ∈ Cat)
7 thincciso.x . . 3 (𝜑𝑋𝐵)
8 thincciso.y . . 3 (𝜑𝑌𝐵)
91, 2, 6, 7, 8cic 17789 . 2 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
10 opex 5470 . . . . . . 7 𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ V
1110a1i 11 . . . . . 6 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ V)
12 biimp 214 . . . . . . . . . . . . 13 (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) → ((𝑥𝐻𝑦) = ∅ → ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅))
13122ralimi 3120 . . . . . . . . . . . 12 (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) → ∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ → ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅))
1413ad2antrl 726 . . . . . . . . . . 11 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → ∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ → ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅))
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 (𝜑𝑌 ∈ ThinCat)
1918adantr 479 . . . . . . . . . . . 12 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑌 ∈ ThinCat)
20 eqid 2728 . . . . . . . . . . . . 13 (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))) = (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))
21 thincciso.s . . . . . . . . . . . . . 14 𝑆 = (Base‘𝑌)
22 thincciso.xt . . . . . . . . . . . . . . . 16 (𝜑𝑋 ∈ ThinCat)
2322adantr 479 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑋 ∈ ThinCat)
2423thinccd 48109 . . . . . . . . . . . . . 14 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑋 ∈ Cat)
25 simprr 771 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑓:𝑅1-1-onto𝑆)
26 f1of 6844 . . . . . . . . . . . . . . 15 (𝑓:𝑅1-1-onto𝑆𝑓:𝑅𝑆)
2725, 26syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑓:𝑅𝑆)
28 biimpr 219 . . . . . . . . . . . . . . . 16 (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) → (((𝑓𝑥)𝐽(𝑓𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
29282ralimi 3120 . . . . . . . . . . . . . . 15 (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) → ∀𝑥𝑅𝑦𝑅 (((𝑓𝑥)𝐽(𝑓𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
3029ad2antrl 726 . . . . . . . . . . . . . 14 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → ∀𝑥𝑅𝑦𝑅 (((𝑓𝑥)𝐽(𝑓𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
3115, 21, 17, 16, 24, 19, 27, 20, 30functhinc 48129 . . . . . . . . . . . . 13 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → (𝑓(𝑋 Func 𝑌)(𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))) ↔ (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))) = (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))))
3220, 31mpbiri 257 . . . . . . . . . . . 12 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑓(𝑋 Func 𝑌)(𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))))
3315, 16, 17, 19, 32fullthinc 48130 . . . . . . . . . . 11 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → (𝑓(𝑋 Full 𝑌)(𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))) ↔ ∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ → ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅)))
3414, 33mpbird 256 . . . . . . . . . 10 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑓(𝑋 Full 𝑌)(𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))))
35 df-br 5153 . . . . . . . . . 10 (𝑓(𝑋 Full 𝑌)(𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))) ↔ ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ (𝑋 Full 𝑌))
3634, 35sylib 217 . . . . . . . . 9 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ (𝑋 Full 𝑌))
3723, 32thincfth 48132 . . . . . . . . . 10 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑓(𝑋 Faith 𝑌)(𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))))
38 df-br 5153 . . . . . . . . . 10 (𝑓(𝑋 Faith 𝑌)(𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))) ↔ ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ (𝑋 Faith 𝑌))
3937, 38sylib 217 . . . . . . . . 9 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ (𝑋 Faith 𝑌))
4036, 39elind 4196 . . . . . . . 8 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
41 vex 3477 . . . . . . . . . . 11 𝑓 ∈ V
4215fvexi 6916 . . . . . . . . . . . 12 𝑅 ∈ V
4342, 42mpoex 8090 . . . . . . . . . . 11 (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))) ∈ V
4441, 43op1st 8007 . . . . . . . . . 10 (1st ‘⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩) = 𝑓
45 f1oeq1 6832 . . . . . . . . . 10 ((1st ‘⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩) = 𝑓 → ((1st ‘⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩):𝑅1-1-onto𝑆𝑓:𝑅1-1-onto𝑆))
4644, 45ax-mp 5 . . . . . . . . 9 ((1st ‘⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩):𝑅1-1-onto𝑆𝑓:𝑅1-1-onto𝑆)
4725, 46sylibr 233 . . . . . . . 8 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → (1st ‘⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩):𝑅1-1-onto𝑆)
4840, 47jca 510 . . . . . . 7 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → (⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩):𝑅1-1-onto𝑆))
494, 2, 15, 21, 3, 7, 8, 1catciso 18107 . . . . . . . 8 (𝜑 → (⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ (𝑋(Iso‘𝐶)𝑌) ↔ (⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩):𝑅1-1-onto𝑆)))
5049biimpar 476 . . . . . . 7 ((𝜑 ∧ (⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st ‘⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩):𝑅1-1-onto𝑆)) → ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ (𝑋(Iso‘𝐶)𝑌))
5148, 50syldan 589 . . . . . 6 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ (𝑋(Iso‘𝐶)𝑌))
52 eleq1 2817 . . . . . 6 (𝑎 = ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ → (𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ (𝑋(Iso‘𝐶)𝑌)))
5311, 51, 52spcedv 3587 . . . . 5 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌))
5453ex 411 . . . 4 (𝜑 → ((∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆) → ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
5554exlimdv 1928 . . 3 (𝜑 → (∃𝑓(∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆) → ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
56 fvexd 6917 . . . . . 6 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st𝑎) ∈ V)
57 relfull 17904 . . . . . . . . . 10 Rel (𝑋 Full 𝑌)
584, 2, 15, 21, 3, 7, 8, 1catciso 18107 . . . . . . . . . . . . 13 (𝜑 → (𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ (𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝑎):𝑅1-1-onto𝑆)))
5958biimpa 475 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝑎):𝑅1-1-onto𝑆))
6059simpld 493 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
6160elin1d 4200 . . . . . . . . . 10 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑎 ∈ (𝑋 Full 𝑌))
62 1st2ndbr 8052 . . . . . . . . . 10 ((Rel (𝑋 Full 𝑌) ∧ 𝑎 ∈ (𝑋 Full 𝑌)) → (1st𝑎)(𝑋 Full 𝑌)(2nd𝑎))
6357, 61, 62sylancr 585 . . . . . . . . 9 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st𝑎)(𝑋 Full 𝑌)(2nd𝑎))
6418adantr 479 . . . . . . . . . 10 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑌 ∈ ThinCat)
65 fullfunc 17902 . . . . . . . . . . . 12 (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
6665ssbri 5197 . . . . . . . . . . 11 ((1st𝑎)(𝑋 Full 𝑌)(2nd𝑎) → (1st𝑎)(𝑋 Func 𝑌)(2nd𝑎))
6763, 66syl 17 . . . . . . . . . 10 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st𝑎)(𝑋 Func 𝑌)(2nd𝑎))
6815, 16, 17, 64, 67fullthinc 48130 . . . . . . . . 9 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ((1st𝑎)(𝑋 Full 𝑌)(2nd𝑎) ↔ ∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ → (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅)))
6963, 68mpbid 231 . . . . . . . 8 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ → (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅))
7067adantr 479 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (1st𝑎)(𝑋 Func 𝑌)(2nd𝑎))
71 simprl 769 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
72 simprr 771 . . . . . . . . . . 11 (((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦𝑅)
7315, 17, 16, 70, 71, 72funcf2 17861 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd𝑎)𝑦):(𝑥𝐻𝑦)⟶(((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)))
7473f002 47984 . . . . . . . . 9 (((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
7574ralrimivva 3198 . . . . . . . 8 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥𝑅𝑦𝑅 ((((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
76 2ralbiim 3129 . . . . . . . 8 (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅) ↔ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ → (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅) ∧ ∀𝑥𝑅𝑦𝑅 ((((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅)))
7769, 75, 76sylanbrc 581 . . . . . . 7 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅))
7859simprd 494 . . . . . . 7 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st𝑎):𝑅1-1-onto𝑆)
7977, 78jca 510 . . . . . 6 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅) ∧ (1st𝑎):𝑅1-1-onto𝑆))
80 fveq1 6901 . . . . . . . . . . 11 (𝑓 = (1st𝑎) → (𝑓𝑥) = ((1st𝑎)‘𝑥))
81 fveq1 6901 . . . . . . . . . . 11 (𝑓 = (1st𝑎) → (𝑓𝑦) = ((1st𝑎)‘𝑦))
8280, 81oveq12d 7444 . . . . . . . . . 10 (𝑓 = (1st𝑎) → ((𝑓𝑥)𝐽(𝑓𝑦)) = (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)))
8382eqeq1d 2730 . . . . . . . . 9 (𝑓 = (1st𝑎) → (((𝑓𝑥)𝐽(𝑓𝑦)) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅))
8483bibi2d 341 . . . . . . . 8 (𝑓 = (1st𝑎) → (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ↔ ((𝑥𝐻𝑦) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅)))
85842ralbidv 3216 . . . . . . 7 (𝑓 = (1st𝑎) → (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ↔ ∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅)))
86 f1oeq1 6832 . . . . . . 7 (𝑓 = (1st𝑎) → (𝑓:𝑅1-1-onto𝑆 ↔ (1st𝑎):𝑅1-1-onto𝑆))
8785, 86anbi12d 630 . . . . . 6 (𝑓 = (1st𝑎) → ((∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆) ↔ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅) ∧ (1st𝑎):𝑅1-1-onto𝑆)))
8856, 79, 87spcedv 3587 . . . . 5 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∃𝑓(∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆))
8988ex 411 . . . 4 (𝜑 → (𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) → ∃𝑓(∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)))
9089exlimdv 1928 . . 3 (𝜑 → (∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) → ∃𝑓(∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)))
9155, 90impbid 211 . 2 (𝜑 → (∃𝑓(∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆) ↔ ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
929, 91bitr4d 281 1 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ ∃𝑓(∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wex 1773  wcel 2098  wral 3058  Vcvv 3473  cin 3948  c0 4326  cop 4638   class class class wbr 5152   × cxp 5680  Rel wrel 5687  wf 6549  1-1-ontowf1o 6552  cfv 6553  (class class class)co 7426  cmpo 7428  1st c1st 7997  2nd c2nd 7998  Basecbs 17187  Hom chom 17251  Catccat 17651  Isociso 17736  𝑐 ccic 17785   Func cfunc 17847   Full cful 17898   Faith cfth 17899  CatCatccatc 18094  ThinCatcthinc 48103
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-tp 4637  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-1st 7999  df-2nd 8000  df-supp 8172  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-1o 8493  df-er 8731  df-map 8853  df-ixp 8923  df-en 8971  df-dom 8972  df-sdom 8973  df-fin 8974  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-9 12320  df-n0 12511  df-z 12597  df-dec 12716  df-uz 12861  df-fz 13525  df-struct 17123  df-slot 17158  df-ndx 17170  df-base 17188  df-hom 17264  df-cco 17265  df-cat 17655  df-cid 17656  df-sect 17737  df-inv 17738  df-iso 17739  df-cic 17786  df-func 17851  df-idfu 17852  df-cofu 17853  df-full 17900  df-fth 17901  df-catc 18095  df-thinc 48104
This theorem is referenced by: (None)
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