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Theorem thincciso 47059
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 2736 . . 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 17994 . . . 4 (𝑈𝑉𝐶 ∈ Cat)
63, 5syl 17 . . 3 (𝜑𝐶 ∈ Cat)
7 thincciso.x . . 3 (𝜑𝑋𝐵)
8 thincciso.y . . 3 (𝜑𝑌𝐵)
91, 2, 6, 7, 8cic 17682 . 2 (𝜑 → (𝑋( ≃𝑐𝐶)𝑌 ↔ ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
10 opex 5421 . . . . . . 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 3126 . . . . . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑌 ∈ ThinCat)
20 eqid 2736 . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑋 ∈ ThinCat)
2423thinccd 47035 . . . . . . . . . . . . . 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 6784 . . . . . . . . . . . . . . 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 3126 . . . . . . . . . . . . . . 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 47055 . . . . . . . . . . . . 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 47056 . . . . . . . . . . 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 5106 . . . . . . . . . 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 47058 . . . . . . . . . 10 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → 𝑓(𝑋 Faith 𝑌)(𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))))
38 df-br 5106 . . . . . . . . . 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 4154 . . . . . . . 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 3449 . . . . . . . . . . 11 𝑓 ∈ V
4215fvexi 6856 . . . . . . . . . . . 12 𝑅 ∈ V
4342, 42mpoex 8012 . . . . . . . . . . 11 (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤)))) ∈ V
4441, 43op1st 7929 . . . . . . . . . 10 (1st ‘⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩) = 𝑓
45 f1oeq1 6772 . . . . . . . . . 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 512 . . . . . . 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 17997 . . . . . . . 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 478 . . . . . . 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 591 . . . . . 6 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → ⟨𝑓, (𝑧𝑅, 𝑤𝑅 ↦ ((𝑧𝐻𝑤) × ((𝑓𝑧)𝐽(𝑓𝑤))))⟩ ∈ (𝑋(Iso‘𝐶)𝑌))
52 eleq1 2825 . . . . . 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 3557 . . . . 5 ((𝜑 ∧ (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)) → ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌))
5453ex 413 . . . 4 (𝜑 → ((∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆) → ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
5554exlimdv 1936 . . 3 (𝜑 → (∃𝑓(∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆) → ∃𝑎 𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)))
56 fvexd 6857 . . . . . 6 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st𝑎) ∈ V)
57 relfull 17795 . . . . . . . . . 10 Rel (𝑋 Full 𝑌)
584, 2, 15, 21, 3, 7, 8, 1catciso 17997 . . . . . . . . . . . . 13 (𝜑 → (𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) ↔ (𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝑎):𝑅1-1-onto𝑆)))
5958biimpa 477 . . . . . . . . . . . 12 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)) ∧ (1st𝑎):𝑅1-1-onto𝑆))
6059simpld 495 . . . . . . . . . . 11 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑎 ∈ ((𝑋 Full 𝑌) ∩ (𝑋 Faith 𝑌)))
6160elin1d 4158 . . . . . . . . . 10 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑎 ∈ (𝑋 Full 𝑌))
62 1st2ndbr 7974 . . . . . . . . . 10 ((Rel (𝑋 Full 𝑌) ∧ 𝑎 ∈ (𝑋 Full 𝑌)) → (1st𝑎)(𝑋 Full 𝑌)(2nd𝑎))
6357, 61, 62sylancr 587 . . . . . . . . 9 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st𝑎)(𝑋 Full 𝑌)(2nd𝑎))
6418adantr 481 . . . . . . . . . 10 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → 𝑌 ∈ ThinCat)
65 fullfunc 17793 . . . . . . . . . . . 12 (𝑋 Full 𝑌) ⊆ (𝑋 Func 𝑌)
6665ssbri 5150 . . . . . . . . . . 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 47056 . . . . . . . . 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 481 . . . . . . . . . . 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 17754 . . . . . . . . . 10 (((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(2nd𝑎)𝑦):(𝑥𝐻𝑦)⟶(((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)))
7473f002 46910 . . . . . . . . 9 (((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) ∧ (𝑥𝑅𝑦𝑅)) → ((((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅ → (𝑥𝐻𝑦) = ∅))
7574ralrimivva 3197 . . . . . . . 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 583 . . . . . . 7 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅))
7859simprd 496 . . . . . . 7 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (1st𝑎):𝑅1-1-onto𝑆)
7977, 78jca 512 . . . . . 6 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅) ∧ (1st𝑎):𝑅1-1-onto𝑆))
80 fveq1 6841 . . . . . . . . . . 11 (𝑓 = (1st𝑎) → (𝑓𝑥) = ((1st𝑎)‘𝑥))
81 fveq1 6841 . . . . . . . . . . 11 (𝑓 = (1st𝑎) → (𝑓𝑦) = ((1st𝑎)‘𝑦))
8280, 81oveq12d 7375 . . . . . . . . . 10 (𝑓 = (1st𝑎) → ((𝑓𝑥)𝐽(𝑓𝑦)) = (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)))
8382eqeq1d 2738 . . . . . . . . 9 (𝑓 = (1st𝑎) → (((𝑓𝑥)𝐽(𝑓𝑦)) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅))
8483bibi2d 342 . . . . . . . 8 (𝑓 = (1st𝑎) → (((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ↔ ((𝑥𝐻𝑦) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅)))
85842ralbidv 3212 . . . . . . 7 (𝑓 = (1st𝑎) → (∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ↔ ∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ (((1st𝑎)‘𝑥)𝐽((1st𝑎)‘𝑦)) = ∅)))
86 f1oeq1 6772 . . . . . . 7 (𝑓 = (1st𝑎) → (𝑓:𝑅1-1-onto𝑆 ↔ (1st𝑎):𝑅1-1-onto𝑆))
8785, 86anbi12d 631 . . . . . 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 3557 . . . . 5 ((𝜑𝑎 ∈ (𝑋(Iso‘𝐶)𝑌)) → ∃𝑓(∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆))
8988ex 413 . . . 4 (𝜑 → (𝑎 ∈ (𝑋(Iso‘𝐶)𝑌) → ∃𝑓(∀𝑥𝑅𝑦𝑅 ((𝑥𝐻𝑦) = ∅ ↔ ((𝑓𝑥)𝐽(𝑓𝑦)) = ∅) ∧ 𝑓:𝑅1-1-onto𝑆)))
9089exlimdv 1936 . . 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 396   = wceq 1541  wex 1781  wcel 2106  wral 3064  Vcvv 3445  cin 3909  c0 4282  cop 4592   class class class wbr 5105   × cxp 5631  Rel wrel 5638  wf 6492  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cmpo 7359  1st c1st 7919  2nd c2nd 7920  Basecbs 17083  Hom chom 17144  Catccat 17544  Isociso 17629  𝑐 ccic 17678   Func cfunc 17740   Full cful 17789   Faith cfth 17790  CatCatccatc 17984  ThinCatcthinc 47029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-hom 17157  df-cco 17158  df-cat 17548  df-cid 17549  df-sect 17630  df-inv 17631  df-iso 17632  df-cic 17679  df-func 17744  df-idfu 17745  df-cofu 17746  df-full 17791  df-fth 17792  df-catc 17985  df-thinc 47030
This theorem is referenced by: (None)
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