Step | Hyp | Ref
| Expression |
1 | | eqid 2733 |
. . 3
β’
(IsoβπΆ) =
(IsoβπΆ) |
2 | | thincciso.b |
. . 3
β’ π΅ = (BaseβπΆ) |
3 | | thincciso.u |
. . . 4
β’ (π β π β π) |
4 | | thincciso.c |
. . . . 5
β’ πΆ = (CatCatβπ) |
5 | 4 | catccat 17999 |
. . . 4
β’ (π β π β πΆ β Cat) |
6 | 3, 5 | syl 17 |
. . 3
β’ (π β πΆ β Cat) |
7 | | thincciso.x |
. . 3
β’ (π β π β π΅) |
8 | | thincciso.y |
. . 3
β’ (π β π β π΅) |
9 | 1, 2, 6, 7, 8 | cic 17687 |
. 2
β’ (π β (π( βπ βπΆ)π β βπ π β (π(IsoβπΆ)π))) |
10 | | opex 5422 |
. . . . . . 7
β’
β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β V |
11 | 10 | a1i 11 |
. . . . . 6
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β V) |
12 | | biimp 214 |
. . . . . . . . . . . . 13
β’ (((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β ((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
)) |
13 | 12 | 2ralimi 3123 |
. . . . . . . . . . . 12
β’
(βπ₯ β
π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
)) |
14 | 13 | ad2antrl 727 |
. . . . . . . . . . 11
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
)) |
15 | | thincciso.r |
. . . . . . . . . . . 12
β’ π
= (Baseβπ) |
16 | | thincciso.j |
. . . . . . . . . . . 12
β’ π½ = (Hom βπ) |
17 | | thincciso.h |
. . . . . . . . . . . 12
β’ π» = (Hom βπ) |
18 | | thincciso.yt |
. . . . . . . . . . . . 13
β’ (π β π β ThinCat) |
19 | 18 | adantr 482 |
. . . . . . . . . . . 12
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β π β ThinCat) |
20 | | eqid 2733 |
. . . . . . . . . . . . 13
β’ (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€)))) = (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€)))) |
21 | | thincciso.s |
. . . . . . . . . . . . . 14
β’ π = (Baseβπ) |
22 | | thincciso.xt |
. . . . . . . . . . . . . . . 16
β’ (π β π β ThinCat) |
23 | 22 | adantr 482 |
. . . . . . . . . . . . . . 15
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β π β ThinCat) |
24 | 23 | thinccd 47131 |
. . . . . . . . . . . . . 14
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β π β Cat) |
25 | | simprr 772 |
. . . . . . . . . . . . . . 15
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β π:π
β1-1-ontoβπ) |
26 | | f1of 6785 |
. . . . . . . . . . . . . . 15
β’ (π:π
β1-1-ontoβπ β π:π
βΆπ) |
27 | 25, 26 | syl 17 |
. . . . . . . . . . . . . 14
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β π:π
βΆπ) |
28 | | biimpr 219 |
. . . . . . . . . . . . . . . 16
β’ (((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β (((πβπ₯)π½(πβπ¦)) = β
β (π₯π»π¦) = β
)) |
29 | 28 | 2ralimi 3123 |
. . . . . . . . . . . . . . 15
β’
(βπ₯ β
π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β βπ₯ β π
βπ¦ β π
(((πβπ₯)π½(πβπ¦)) = β
β (π₯π»π¦) = β
)) |
30 | 29 | ad2antrl 727 |
. . . . . . . . . . . . . 14
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β βπ₯ β π
βπ¦ β π
(((πβπ₯)π½(πβπ¦)) = β
β (π₯π»π¦) = β
)) |
31 | 15, 21, 17, 16, 24, 19, 27, 20, 30 | functhinc 47151 |
. . . . . . . . . . . . 13
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β (π(π Func π)(π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€)))) β (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€)))) = (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€)))))) |
32 | 20, 31 | mpbiri 258 |
. . . . . . . . . . . 12
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β π(π Func π)(π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))) |
33 | 15, 16, 17, 19, 32 | fullthinc 47152 |
. . . . . . . . . . 11
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β (π(π Full π)(π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€)))) β βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
))) |
34 | 14, 33 | mpbird 257 |
. . . . . . . . . 10
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β π(π Full π)(π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))) |
35 | | df-br 5107 |
. . . . . . . . . 10
β’ (π(π Full π)(π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€)))) β β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β (π Full π)) |
36 | 34, 35 | sylib 217 |
. . . . . . . . 9
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β (π Full π)) |
37 | 23, 32 | thincfth 47154 |
. . . . . . . . . 10
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β π(π Faith π)(π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))) |
38 | | df-br 5107 |
. . . . . . . . . 10
β’ (π(π Faith π)(π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€)))) β β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β (π Faith π)) |
39 | 37, 38 | sylib 217 |
. . . . . . . . 9
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β (π Faith π)) |
40 | 36, 39 | elind 4155 |
. . . . . . . 8
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β ((π Full π) β© (π Faith π))) |
41 | | vex 3448 |
. . . . . . . . . . 11
β’ π β V |
42 | 15 | fvexi 6857 |
. . . . . . . . . . . 12
β’ π
β V |
43 | 42, 42 | mpoex 8013 |
. . . . . . . . . . 11
β’ (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€)))) β V |
44 | 41, 43 | op1st 7930 |
. . . . . . . . . 10
β’
(1st ββ¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β©) = π |
45 | | f1oeq1 6773 |
. . . . . . . . . 10
β’
((1st ββ¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β©) = π β ((1st ββ¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β©):π
β1-1-ontoβπ β π:π
β1-1-ontoβπ)) |
46 | 44, 45 | ax-mp 5 |
. . . . . . . . 9
β’
((1st ββ¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β©):π
β1-1-ontoβπ β π:π
β1-1-ontoβπ) |
47 | 25, 46 | sylibr 233 |
. . . . . . . 8
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β (1st
ββ¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β©):π
β1-1-ontoβπ) |
48 | 40, 47 | jca 513 |
. . . . . . 7
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β (β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β ((π Full π) β© (π Faith π)) β§ (1st ββ¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β©):π
β1-1-ontoβπ)) |
49 | 4, 2, 15, 21, 3, 7, 8, 1 | catciso 18002 |
. . . . . . . 8
β’ (π β (β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β (π(IsoβπΆ)π) β (β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β ((π Full π) β© (π Faith π)) β§ (1st ββ¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β©):π
β1-1-ontoβπ))) |
50 | 49 | biimpar 479 |
. . . . . . 7
β’ ((π β§ (β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β ((π Full π) β© (π Faith π)) β§ (1st ββ¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β©):π
β1-1-ontoβπ)) β β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β (π(IsoβπΆ)π)) |
51 | 48, 50 | syldan 592 |
. . . . . 6
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β (π(IsoβπΆ)π)) |
52 | | eleq1 2822 |
. . . . . 6
β’ (π = β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β (π β (π(IsoβπΆ)π) β β¨π, (π§ β π
, π€ β π
β¦ ((π§π»π€) Γ ((πβπ§)π½(πβπ€))))β© β (π(IsoβπΆ)π))) |
53 | 11, 51, 52 | spcedv 3556 |
. . . . 5
β’ ((π β§ (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) β βπ π β (π(IsoβπΆ)π)) |
54 | 53 | ex 414 |
. . . 4
β’ (π β ((βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ) β βπ π β (π(IsoβπΆ)π))) |
55 | 54 | exlimdv 1937 |
. . 3
β’ (π β (βπ(βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ) β βπ π β (π(IsoβπΆ)π))) |
56 | | fvexd 6858 |
. . . . . 6
β’ ((π β§ π β (π(IsoβπΆ)π)) β (1st βπ) β V) |
57 | | relfull 17800 |
. . . . . . . . . 10
β’ Rel
(π Full π) |
58 | 4, 2, 15, 21, 3, 7, 8, 1 | catciso 18002 |
. . . . . . . . . . . . 13
β’ (π β (π β (π(IsoβπΆ)π) β (π β ((π Full π) β© (π Faith π)) β§ (1st βπ):π
β1-1-ontoβπ))) |
59 | 58 | biimpa 478 |
. . . . . . . . . . . 12
β’ ((π β§ π β (π(IsoβπΆ)π)) β (π β ((π Full π) β© (π Faith π)) β§ (1st βπ):π
β1-1-ontoβπ)) |
60 | 59 | simpld 496 |
. . . . . . . . . . 11
β’ ((π β§ π β (π(IsoβπΆ)π)) β π β ((π Full π) β© (π Faith π))) |
61 | 60 | elin1d 4159 |
. . . . . . . . . 10
β’ ((π β§ π β (π(IsoβπΆ)π)) β π β (π Full π)) |
62 | | 1st2ndbr 7975 |
. . . . . . . . . 10
β’ ((Rel
(π Full π) β§ π β (π Full π)) β (1st βπ)(π Full π)(2nd βπ)) |
63 | 57, 61, 62 | sylancr 588 |
. . . . . . . . 9
β’ ((π β§ π β (π(IsoβπΆ)π)) β (1st βπ)(π Full π)(2nd βπ)) |
64 | 18 | adantr 482 |
. . . . . . . . . 10
β’ ((π β§ π β (π(IsoβπΆ)π)) β π β ThinCat) |
65 | | fullfunc 17798 |
. . . . . . . . . . . 12
β’ (π Full π) β (π Func π) |
66 | 65 | ssbri 5151 |
. . . . . . . . . . 11
β’
((1st βπ)(π Full π)(2nd βπ) β (1st βπ)(π Func π)(2nd βπ)) |
67 | 63, 66 | syl 17 |
. . . . . . . . . 10
β’ ((π β§ π β (π(IsoβπΆ)π)) β (1st βπ)(π Func π)(2nd βπ)) |
68 | 15, 16, 17, 64, 67 | fullthinc 47152 |
. . . . . . . . 9
β’ ((π β§ π β (π(IsoβπΆ)π)) β ((1st βπ)(π Full π)(2nd βπ) β βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β (((1st
βπ)βπ₯)π½((1st βπ)βπ¦)) = β
))) |
69 | 63, 68 | mpbid 231 |
. . . . . . . 8
β’ ((π β§ π β (π(IsoβπΆ)π)) β βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β (((1st
βπ)βπ₯)π½((1st βπ)βπ¦)) = β
)) |
70 | 67 | adantr 482 |
. . . . . . . . . . 11
β’ (((π β§ π β (π(IsoβπΆ)π)) β§ (π₯ β π
β§ π¦ β π
)) β (1st βπ)(π Func π)(2nd βπ)) |
71 | | simprl 770 |
. . . . . . . . . . 11
β’ (((π β§ π β (π(IsoβπΆ)π)) β§ (π₯ β π
β§ π¦ β π
)) β π₯ β π
) |
72 | | simprr 772 |
. . . . . . . . . . 11
β’ (((π β§ π β (π(IsoβπΆ)π)) β§ (π₯ β π
β§ π¦ β π
)) β π¦ β π
) |
73 | 15, 17, 16, 70, 71, 72 | funcf2 17759 |
. . . . . . . . . 10
β’ (((π β§ π β (π(IsoβπΆ)π)) β§ (π₯ β π
β§ π¦ β π
)) β (π₯(2nd βπ)π¦):(π₯π»π¦)βΆ(((1st βπ)βπ₯)π½((1st βπ)βπ¦))) |
74 | 73 | f002 47006 |
. . . . . . . . 9
β’ (((π β§ π β (π(IsoβπΆ)π)) β§ (π₯ β π
β§ π¦ β π
)) β ((((1st βπ)βπ₯)π½((1st βπ)βπ¦)) = β
β (π₯π»π¦) = β
)) |
75 | 74 | ralrimivva 3194 |
. . . . . . . 8
β’ ((π β§ π β (π(IsoβπΆ)π)) β βπ₯ β π
βπ¦ β π
((((1st βπ)βπ₯)π½((1st βπ)βπ¦)) = β
β (π₯π»π¦) = β
)) |
76 | | 2ralbiim 3126 |
. . . . . . . 8
β’
(βπ₯ β
π
βπ¦ β π
((π₯π»π¦) = β
β (((1st
βπ)βπ₯)π½((1st βπ)βπ¦)) = β
) β (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β (((1st
βπ)βπ₯)π½((1st βπ)βπ¦)) = β
) β§ βπ₯ β π
βπ¦ β π
((((1st βπ)βπ₯)π½((1st βπ)βπ¦)) = β
β (π₯π»π¦) = β
))) |
77 | 69, 75, 76 | sylanbrc 584 |
. . . . . . 7
β’ ((π β§ π β (π(IsoβπΆ)π)) β βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β (((1st
βπ)βπ₯)π½((1st βπ)βπ¦)) = β
)) |
78 | 59 | simprd 497 |
. . . . . . 7
β’ ((π β§ π β (π(IsoβπΆ)π)) β (1st βπ):π
β1-1-ontoβπ) |
79 | 77, 78 | jca 513 |
. . . . . 6
β’ ((π β§ π β (π(IsoβπΆ)π)) β (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β (((1st
βπ)βπ₯)π½((1st βπ)βπ¦)) = β
) β§ (1st
βπ):π
β1-1-ontoβπ)) |
80 | | fveq1 6842 |
. . . . . . . . . . 11
β’ (π = (1st βπ) β (πβπ₯) = ((1st βπ)βπ₯)) |
81 | | fveq1 6842 |
. . . . . . . . . . 11
β’ (π = (1st βπ) β (πβπ¦) = ((1st βπ)βπ¦)) |
82 | 80, 81 | oveq12d 7376 |
. . . . . . . . . 10
β’ (π = (1st βπ) β ((πβπ₯)π½(πβπ¦)) = (((1st βπ)βπ₯)π½((1st βπ)βπ¦))) |
83 | 82 | eqeq1d 2735 |
. . . . . . . . 9
β’ (π = (1st βπ) β (((πβπ₯)π½(πβπ¦)) = β
β (((1st
βπ)βπ₯)π½((1st βπ)βπ¦)) = β
)) |
84 | 83 | bibi2d 343 |
. . . . . . . 8
β’ (π = (1st βπ) β (((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β ((π₯π»π¦) = β
β (((1st
βπ)βπ₯)π½((1st βπ)βπ¦)) = β
))) |
85 | 84 | 2ralbidv 3209 |
. . . . . . 7
β’ (π = (1st βπ) β (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β (((1st
βπ)βπ₯)π½((1st βπ)βπ¦)) = β
))) |
86 | | f1oeq1 6773 |
. . . . . . 7
β’ (π = (1st βπ) β (π:π
β1-1-ontoβπ β (1st
βπ):π
β1-1-ontoβπ)) |
87 | 85, 86 | anbi12d 632 |
. . . . . 6
β’ (π = (1st βπ) β ((βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ) β (βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β (((1st
βπ)βπ₯)π½((1st βπ)βπ¦)) = β
) β§ (1st
βπ):π
β1-1-ontoβπ))) |
88 | 56, 79, 87 | spcedv 3556 |
. . . . 5
β’ ((π β§ π β (π(IsoβπΆ)π)) β βπ(βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ)) |
89 | 88 | ex 414 |
. . . 4
β’ (π β (π β (π(IsoβπΆ)π) β βπ(βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ))) |
90 | 89 | exlimdv 1937 |
. . 3
β’ (π β (βπ π β (π(IsoβπΆ)π) β βπ(βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ))) |
91 | 55, 90 | impbid 211 |
. 2
β’ (π β (βπ(βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ) β βπ π β (π(IsoβπΆ)π))) |
92 | 9, 91 | bitr4d 282 |
1
β’ (π β (π( βπ βπΆ)π β βπ(βπ₯ β π
βπ¦ β π
((π₯π»π¦) = β
β ((πβπ₯)π½(πβπ¦)) = β
) β§ π:π
β1-1-ontoβπ))) |