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Theorem thincciso 47155
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 𝐢 = (CatCatβ€˜π‘ˆ)
thincciso.b 𝐡 = (Baseβ€˜πΆ)
thincciso.r 𝑅 = (Baseβ€˜π‘‹)
thincciso.s 𝑆 = (Baseβ€˜π‘Œ)
thincciso.h 𝐻 = (Hom β€˜π‘‹)
thincciso.j 𝐽 = (Hom β€˜π‘Œ)
thincciso.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
thincciso.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
thincciso.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
thincciso.xt (πœ‘ β†’ 𝑋 ∈ ThinCat)
thincciso.yt (πœ‘ β†’ π‘Œ ∈ ThinCat)
Assertion
Ref Expression
thincciso (πœ‘ β†’ (𝑋( ≃𝑐 β€˜πΆ)π‘Œ ↔ βˆƒπ‘“(βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)))
Distinct variable groups:   𝐢,𝑓,π‘₯,𝑦   𝑓,𝐻,π‘₯,𝑦   𝑓,𝐽,π‘₯,𝑦   𝑅,𝑓,π‘₯,𝑦   𝑆,𝑓   𝑓,𝑋,π‘₯,𝑦   𝑓,π‘Œ,π‘₯,𝑦   πœ‘,𝑓,π‘₯,𝑦
Allowed substitution hints:   𝐡(π‘₯,𝑦,𝑓)   𝑆(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦,𝑓)   𝑉(π‘₯,𝑦,𝑓)

Proof of Theorem thincciso
Dummy variables π‘Ž 𝑀 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . 3 (Isoβ€˜πΆ) = (Isoβ€˜πΆ)
2 thincciso.b . . 3 𝐡 = (Baseβ€˜πΆ)
3 thincciso.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
4 thincciso.c . . . . 5 𝐢 = (CatCatβ€˜π‘ˆ)
54catccat 17999 . . . 4 (π‘ˆ ∈ 𝑉 β†’ 𝐢 ∈ Cat)
63, 5syl 17 . . 3 (πœ‘ β†’ 𝐢 ∈ Cat)
7 thincciso.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
8 thincciso.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
91, 2, 6, 7, 8cic 17687 . 2 (πœ‘ β†’ (𝑋( ≃𝑐 β€˜πΆ)π‘Œ ↔ βˆƒπ‘Ž π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)))
10 opex 5422 . . . . . . 7 βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ V
1110a1i 11 . . . . . 6 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ V)
12 biimp 214 . . . . . . . . . . . . 13 (((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) β†’ ((π‘₯𝐻𝑦) = βˆ… β†’ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…))
13122ralimi 3123 . . . . . . . . . . . 12 (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) β†’ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… β†’ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…))
1413ad2antrl 727 . . . . . . . . . . 11 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… β†’ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…))
15 thincciso.r . . . . . . . . . . . 12 𝑅 = (Baseβ€˜π‘‹)
16 thincciso.j . . . . . . . . . . . 12 𝐽 = (Hom β€˜π‘Œ)
17 thincciso.h . . . . . . . . . . . 12 𝐻 = (Hom β€˜π‘‹)
18 thincciso.yt . . . . . . . . . . . . 13 (πœ‘ β†’ π‘Œ ∈ ThinCat)
1918adantr 482 . . . . . . . . . . . 12 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ π‘Œ ∈ ThinCat)
20 eqid 2733 . . . . . . . . . . . . 13 (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))) = (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))
21 thincciso.s . . . . . . . . . . . . . 14 𝑆 = (Baseβ€˜π‘Œ)
22 thincciso.xt . . . . . . . . . . . . . . . 16 (πœ‘ β†’ 𝑋 ∈ ThinCat)
2322adantr 482 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ 𝑋 ∈ ThinCat)
2423thinccd 47131 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ 𝑋 ∈ Cat)
25 simprr 772 . . . . . . . . . . . . . . 15 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ 𝑓:𝑅–1-1-onto→𝑆)
26 f1of 6785 . . . . . . . . . . . . . . 15 (𝑓:𝑅–1-1-onto→𝑆 β†’ 𝑓:π‘…βŸΆπ‘†)
2725, 26syl 17 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ 𝑓:π‘…βŸΆπ‘†)
28 biimpr 219 . . . . . . . . . . . . . . . 16 (((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) β†’ (((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ… β†’ (π‘₯𝐻𝑦) = βˆ…))
29282ralimi 3123 . . . . . . . . . . . . . . 15 (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) β†’ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 (((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ… β†’ (π‘₯𝐻𝑦) = βˆ…))
3029ad2antrl 727 . . . . . . . . . . . . . 14 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 (((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ… β†’ (π‘₯𝐻𝑦) = βˆ…))
3115, 21, 17, 16, 24, 19, 27, 20, 30functhinc 47151 . . . . . . . . . . . . 13 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ (𝑓(𝑋 Func π‘Œ)(𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))) ↔ (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))) = (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))))
3220, 31mpbiri 258 . . . . . . . . . . . 12 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ 𝑓(𝑋 Func π‘Œ)(𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))))
3315, 16, 17, 19, 32fullthinc 47152 . . . . . . . . . . 11 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ (𝑓(𝑋 Full π‘Œ)(𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))) ↔ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… β†’ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…)))
3414, 33mpbird 257 . . . . . . . . . 10 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ 𝑓(𝑋 Full π‘Œ)(𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))))
35 df-br 5107 . . . . . . . . . 10 (𝑓(𝑋 Full π‘Œ)(𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))) ↔ βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ (𝑋 Full π‘Œ))
3634, 35sylib 217 . . . . . . . . 9 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ (𝑋 Full π‘Œ))
3723, 32thincfth 47154 . . . . . . . . . 10 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ 𝑓(𝑋 Faith π‘Œ)(𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))))
38 df-br 5107 . . . . . . . . . 10 (𝑓(𝑋 Faith π‘Œ)(𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))) ↔ βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ (𝑋 Faith π‘Œ))
3937, 38sylib 217 . . . . . . . . 9 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ (𝑋 Faith π‘Œ))
4036, 39elind 4155 . . . . . . . 8 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ ((𝑋 Full π‘Œ) ∩ (𝑋 Faith π‘Œ)))
41 vex 3448 . . . . . . . . . . 11 𝑓 ∈ V
4215fvexi 6857 . . . . . . . . . . . 12 𝑅 ∈ V
4342, 42mpoex 8013 . . . . . . . . . . 11 (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))) ∈ V
4441, 43op1st 7930 . . . . . . . . . 10 (1st β€˜βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩) = 𝑓
45 f1oeq1 6773 . . . . . . . . . 10 ((1st β€˜βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩) = 𝑓 β†’ ((1st β€˜βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩):𝑅–1-1-onto→𝑆 ↔ 𝑓:𝑅–1-1-onto→𝑆))
4644, 45ax-mp 5 . . . . . . . . 9 ((1st β€˜βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩):𝑅–1-1-onto→𝑆 ↔ 𝑓:𝑅–1-1-onto→𝑆)
4725, 46sylibr 233 . . . . . . . 8 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ (1st β€˜βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩):𝑅–1-1-onto→𝑆)
4840, 47jca 513 . . . . . . 7 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ (βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ ((𝑋 Full π‘Œ) ∩ (𝑋 Faith π‘Œ)) ∧ (1st β€˜βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩):𝑅–1-1-onto→𝑆))
494, 2, 15, 21, 3, 7, 8, 1catciso 18002 . . . . . . . 8 (πœ‘ β†’ (βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ (𝑋(Isoβ€˜πΆ)π‘Œ) ↔ (βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ ((𝑋 Full π‘Œ) ∩ (𝑋 Faith π‘Œ)) ∧ (1st β€˜βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩):𝑅–1-1-onto→𝑆)))
5049biimpar 479 . . . . . . 7 ((πœ‘ ∧ (βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ ((𝑋 Full π‘Œ) ∩ (𝑋 Faith π‘Œ)) ∧ (1st β€˜βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩):𝑅–1-1-onto→𝑆)) β†’ βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ (𝑋(Isoβ€˜πΆ)π‘Œ))
5148, 50syldan 592 . . . . . 6 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ (𝑋(Isoβ€˜πΆ)π‘Œ))
52 eleq1 2822 . . . . . 6 (π‘Ž = βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ β†’ (π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ) ↔ βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩ ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)))
5311, 51, 52spcedv 3556 . . . . 5 ((πœ‘ ∧ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)) β†’ βˆƒπ‘Ž π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ))
5453ex 414 . . . 4 (πœ‘ β†’ ((βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆) β†’ βˆƒπ‘Ž π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)))
5554exlimdv 1937 . . 3 (πœ‘ β†’ (βˆƒπ‘“(βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆) β†’ βˆƒπ‘Ž π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)))
56 fvexd 6858 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ (1st β€˜π‘Ž) ∈ V)
57 relfull 17800 . . . . . . . . . 10 Rel (𝑋 Full π‘Œ)
584, 2, 15, 21, 3, 7, 8, 1catciso 18002 . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ) ↔ (π‘Ž ∈ ((𝑋 Full π‘Œ) ∩ (𝑋 Faith π‘Œ)) ∧ (1st β€˜π‘Ž):𝑅–1-1-onto→𝑆)))
5958biimpa 478 . . . . . . . . . . . 12 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ (π‘Ž ∈ ((𝑋 Full π‘Œ) ∩ (𝑋 Faith π‘Œ)) ∧ (1st β€˜π‘Ž):𝑅–1-1-onto→𝑆))
6059simpld 496 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ π‘Ž ∈ ((𝑋 Full π‘Œ) ∩ (𝑋 Faith π‘Œ)))
6160elin1d 4159 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ π‘Ž ∈ (𝑋 Full π‘Œ))
62 1st2ndbr 7975 . . . . . . . . . 10 ((Rel (𝑋 Full π‘Œ) ∧ π‘Ž ∈ (𝑋 Full π‘Œ)) β†’ (1st β€˜π‘Ž)(𝑋 Full π‘Œ)(2nd β€˜π‘Ž))
6357, 61, 62sylancr 588 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ (1st β€˜π‘Ž)(𝑋 Full π‘Œ)(2nd β€˜π‘Ž))
6418adantr 482 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ π‘Œ ∈ ThinCat)
65 fullfunc 17798 . . . . . . . . . . . 12 (𝑋 Full π‘Œ) βŠ† (𝑋 Func π‘Œ)
6665ssbri 5151 . . . . . . . . . . 11 ((1st β€˜π‘Ž)(𝑋 Full π‘Œ)(2nd β€˜π‘Ž) β†’ (1st β€˜π‘Ž)(𝑋 Func π‘Œ)(2nd β€˜π‘Ž))
6763, 66syl 17 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ (1st β€˜π‘Ž)(𝑋 Func π‘Œ)(2nd β€˜π‘Ž))
6815, 16, 17, 64, 67fullthinc 47152 . . . . . . . . 9 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ ((1st β€˜π‘Ž)(𝑋 Full π‘Œ)(2nd β€˜π‘Ž) ↔ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… β†’ (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ…)))
6963, 68mpbid 231 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… β†’ (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ…))
7067adantr 482 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) ∧ (π‘₯ ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) β†’ (1st β€˜π‘Ž)(𝑋 Func π‘Œ)(2nd β€˜π‘Ž))
71 simprl 770 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) ∧ (π‘₯ ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) β†’ π‘₯ ∈ 𝑅)
72 simprr 772 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) ∧ (π‘₯ ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) β†’ 𝑦 ∈ 𝑅)
7315, 17, 16, 70, 71, 72funcf2 17759 . . . . . . . . . 10 (((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) ∧ (π‘₯ ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) β†’ (π‘₯(2nd β€˜π‘Ž)𝑦):(π‘₯𝐻𝑦)⟢(((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)))
7473f002 47006 . . . . . . . . 9 (((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) ∧ (π‘₯ ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) β†’ ((((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ… β†’ (π‘₯𝐻𝑦) = βˆ…))
7574ralrimivva 3194 . . . . . . . 8 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ… β†’ (π‘₯𝐻𝑦) = βˆ…))
76 2ralbiim 3126 . . . . . . . 8 (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ…) ↔ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… β†’ (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ…) ∧ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ… β†’ (π‘₯𝐻𝑦) = βˆ…)))
7769, 75, 76sylanbrc 584 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ…))
7859simprd 497 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ (1st β€˜π‘Ž):𝑅–1-1-onto→𝑆)
7977, 78jca 513 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ…) ∧ (1st β€˜π‘Ž):𝑅–1-1-onto→𝑆))
80 fveq1 6842 . . . . . . . . . . 11 (𝑓 = (1st β€˜π‘Ž) β†’ (π‘“β€˜π‘₯) = ((1st β€˜π‘Ž)β€˜π‘₯))
81 fveq1 6842 . . . . . . . . . . 11 (𝑓 = (1st β€˜π‘Ž) β†’ (π‘“β€˜π‘¦) = ((1st β€˜π‘Ž)β€˜π‘¦))
8280, 81oveq12d 7376 . . . . . . . . . 10 (𝑓 = (1st β€˜π‘Ž) β†’ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)))
8382eqeq1d 2735 . . . . . . . . 9 (𝑓 = (1st β€˜π‘Ž) β†’ (((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ… ↔ (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ…))
8483bibi2d 343 . . . . . . . 8 (𝑓 = (1st β€˜π‘Ž) β†’ (((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ↔ ((π‘₯𝐻𝑦) = βˆ… ↔ (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ…)))
85842ralbidv 3209 . . . . . . 7 (𝑓 = (1st β€˜π‘Ž) β†’ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ↔ βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ…)))
86 f1oeq1 6773 . . . . . . 7 (𝑓 = (1st β€˜π‘Ž) β†’ (𝑓:𝑅–1-1-onto→𝑆 ↔ (1st β€˜π‘Ž):𝑅–1-1-onto→𝑆))
8785, 86anbi12d 632 . . . . . 6 (𝑓 = (1st β€˜π‘Ž) β†’ ((βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆) ↔ (βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ (((1st β€˜π‘Ž)β€˜π‘₯)𝐽((1st β€˜π‘Ž)β€˜π‘¦)) = βˆ…) ∧ (1st β€˜π‘Ž):𝑅–1-1-onto→𝑆)))
8856, 79, 87spcedv 3556 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ βˆƒπ‘“(βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆))
8988ex 414 . . . 4 (πœ‘ β†’ (π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ) β†’ βˆƒπ‘“(βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)))
9089exlimdv 1937 . . 3 (πœ‘ β†’ (βˆƒπ‘Ž π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ) β†’ βˆƒπ‘“(βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)))
9155, 90impbid 211 . 2 (πœ‘ β†’ (βˆƒπ‘“(βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆) ↔ βˆƒπ‘Ž π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)))
929, 91bitr4d 282 1 (πœ‘ β†’ (𝑋( ≃𝑐 β€˜πΆ)π‘Œ ↔ βˆƒπ‘“(βˆ€π‘₯ ∈ 𝑅 βˆ€π‘¦ ∈ 𝑅 ((π‘₯𝐻𝑦) = βˆ… ↔ ((π‘“β€˜π‘₯)𝐽(π‘“β€˜π‘¦)) = βˆ…) ∧ 𝑓:𝑅–1-1-onto→𝑆)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444   ∩ cin 3910  βˆ…c0 4283  βŸ¨cop 4593   class class class wbr 5106   Γ— cxp 5632  Rel wrel 5639  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921  Basecbs 17088  Hom chom 17149  Catccat 17549  Isociso 17634   ≃𝑐 ccic 17683   Func cfunc 17745   Full cful 17794   Faith cfth 17795  CatCatccatc 17989  ThinCatcthinc 47125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-supp 8094  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8651  df-map 8770  df-ixp 8839  df-en 8887  df-dom 8888  df-sdom 8889  df-fin 8890  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-3 12222  df-4 12223  df-5 12224  df-6 12225  df-7 12226  df-8 12227  df-9 12228  df-n0 12419  df-z 12505  df-dec 12624  df-uz 12769  df-fz 13431  df-struct 17024  df-slot 17059  df-ndx 17071  df-base 17089  df-hom 17162  df-cco 17163  df-cat 17553  df-cid 17554  df-sect 17635  df-inv 17636  df-iso 17637  df-cic 17684  df-func 17749  df-idfu 17750  df-cofu 17751  df-full 17796  df-fth 17797  df-catc 17990  df-thinc 47126
This theorem is referenced by: (None)
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