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Theorem thincciso 48167
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 2725 . . 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 18096 . . . 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 17781 . 2 (πœ‘ β†’ (𝑋( ≃𝑐 β€˜πΆ)π‘Œ ↔ βˆƒπ‘Ž π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)))
10 opex 5460 . . . . . . 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 3113 . . . . . . . . . . . 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 726 . . . . . . . . . . 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 479 . . . . . . . . . . . 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 2725 . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . 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 48143 . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . 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 3113 . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . 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 48163 . . . . . . . . . . . . 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 257 . . . . . . . . . . . 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 48164 . . . . . . . . . . 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 256 . . . . . . . . . 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 5144 . . . . . . . . . 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 48166 . . . . . . . . . 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 5144 . . . . . . . . . 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 4188 . . . . . . . 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 3467 . . . . . . . . . . 11 𝑓 ∈ V
4215fvexi 6906 . . . . . . . . . . . 12 𝑅 ∈ V
4342, 42mpoex 8082 . . . . . . . . . . 11 (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€)))) ∈ V
4441, 43op1st 7999 . . . . . . . . . 10 (1st β€˜βŸ¨π‘“, (𝑧 ∈ 𝑅, 𝑀 ∈ 𝑅 ↦ ((𝑧𝐻𝑀) Γ— ((π‘“β€˜π‘§)𝐽(π‘“β€˜π‘€))))⟩) = 𝑓
45 f1oeq1 6822 . . . . . . . . . 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 510 . . . . . . 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 18099 . . . . . . . 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 476 . . . . . . 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 589 . . . . . 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 2813 . . . . . 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 3577 . . . . 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 411 . . . 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 1928 . . 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 6907 . . . . . 6 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ (1st β€˜π‘Ž) ∈ V)
57 relfull 17896 . . . . . . . . . 10 Rel (𝑋 Full π‘Œ)
584, 2, 15, 21, 3, 7, 8, 1catciso 18099 . . . . . . . . . . . . 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 475 . . . . . . . . . . . 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 493 . . . . . . . . . . 11 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ π‘Ž ∈ ((𝑋 Full π‘Œ) ∩ (𝑋 Faith π‘Œ)))
6160elin1d 4192 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ π‘Ž ∈ (𝑋 Full π‘Œ))
62 1st2ndbr 8044 . . . . . . . . . 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 585 . . . . . . . . 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 479 . . . . . . . . . 10 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ π‘Œ ∈ ThinCat)
65 fullfunc 17894 . . . . . . . . . . . 12 (𝑋 Full π‘Œ) βŠ† (𝑋 Func π‘Œ)
6665ssbri 5188 . . . . . . . . . . 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 48164 . . . . . . . . 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 479 . . . . . . . . . . 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 769 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) ∧ (π‘₯ ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) β†’ π‘₯ ∈ 𝑅)
72 simprr 771 . . . . . . . . . . 11 (((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) ∧ (π‘₯ ∈ 𝑅 ∧ 𝑦 ∈ 𝑅)) β†’ 𝑦 ∈ 𝑅)
7315, 17, 16, 70, 71, 72funcf2 17853 . . . . . . . . . 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 48018 . . . . . . . . 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 3191 . . . . . . . 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 3122 . . . . . . . 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 581 . . . . . . 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 494 . . . . . . 7 ((πœ‘ ∧ π‘Ž ∈ (𝑋(Isoβ€˜πΆ)π‘Œ)) β†’ (1st β€˜π‘Ž):𝑅–1-1-onto→𝑆)
7977, 78jca 510 . . . . . 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 6891 . . . . . . . . . . 11 (𝑓 = (1st β€˜π‘Ž) β†’ (π‘“β€˜π‘₯) = ((1st β€˜π‘Ž)β€˜π‘₯))
81 fveq1 6891 . . . . . . . . . . 11 (𝑓 = (1st β€˜π‘Ž) β†’ (π‘“β€˜π‘¦) = ((1st β€˜π‘Ž)β€˜π‘¦))
8280, 81oveq12d 7434 . . . . . . . . . 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 2727 . . . . . . . . 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 341 . . . . . . . 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 6822 . . . . . . 7 (𝑓 = (1st β€˜π‘Ž) β†’ (𝑓:𝑅–1-1-onto→𝑆 ↔ (1st β€˜π‘Ž):𝑅–1-1-onto→𝑆))
8785, 86anbi12d 630 . . . . . 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 3577 . . . . 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 411 . . . 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 1928 . . 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 281 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 394   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  βˆ€wral 3051  Vcvv 3463   ∩ cin 3938  βˆ…c0 4318  βŸ¨cop 4630   class class class wbr 5143   Γ— cxp 5670  Rel wrel 5677  βŸΆwf 6539  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7416   ∈ cmpo 7418  1st c1st 7989  2nd c2nd 7990  Basecbs 17179  Hom chom 17243  Catccat 17643  Isociso 17728   ≃𝑐 ccic 17777   Func cfunc 17839   Full cful 17890   Faith cfth 17891  CatCatccatc 18086  ThinCatcthinc 48137
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 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-supp 8164  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-ixp 8915  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12503  df-z 12589  df-dec 12708  df-uz 12853  df-fz 13517  df-struct 17115  df-slot 17150  df-ndx 17162  df-base 17180  df-hom 17256  df-cco 17257  df-cat 17647  df-cid 17648  df-sect 17729  df-inv 17730  df-iso 17731  df-cic 17778  df-func 17843  df-idfu 17844  df-cofu 17845  df-full 17892  df-fth 17893  df-catc 18087  df-thinc 48138
This theorem is referenced by: (None)
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