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Theorem fuciso 15466
Description: A natural transformation is an isomorphism of functors iff all its components are isomorphisms. (Contributed by Mario Carneiro, 28-Jan-2017.)
Hypotheses
Ref Expression
fuciso.q  |-  Q  =  ( C FuncCat  D )
fuciso.b  |-  B  =  ( Base `  C
)
fuciso.n  |-  N  =  ( C Nat  D )
fuciso.f  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
fuciso.g  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
fuciso.i  |-  I  =  (  Iso  `  Q
)
fuciso.j  |-  J  =  (  Iso  `  D
)
Assertion
Ref Expression
fuciso  |-  ( ph  ->  ( A  e.  ( F I G )  <-> 
( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, D    x, I    x, F   
x, G    x, J    x, N    ph, x    x, Q

Proof of Theorem fuciso
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fuciso.q . . . . . 6  |-  Q  =  ( C FuncCat  D )
21fucbas 15451 . . . . 5  |-  ( C 
Func  D )  =  (
Base `  Q )
3 fuciso.n . . . . . 6  |-  N  =  ( C Nat  D )
41, 3fuchom 15452 . . . . 5  |-  N  =  ( Hom  `  Q
)
5 fuciso.i . . . . 5  |-  I  =  (  Iso  `  Q
)
6 fuciso.f . . . . . . . 8  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
7 funcrcl 15354 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
86, 7syl 16 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
98simpld 457 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
108simprd 461 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
111, 9, 10fuccat 15461 . . . . 5  |-  ( ph  ->  Q  e.  Cat )
12 fuciso.g . . . . 5  |-  ( ph  ->  G  e.  ( C 
Func  D ) )
132, 4, 5, 11, 6, 12isohom 15267 . . . 4  |-  ( ph  ->  ( F I G )  C_  ( F N G ) )
1413sselda 3489 . . 3  |-  ( (
ph  /\  A  e.  ( F I G ) )  ->  A  e.  ( F N G ) )
15 eqid 2454 . . . . 5  |-  ( Base `  D )  =  (
Base `  D )
16 eqid 2454 . . . . 5  |-  (Inv `  D )  =  (Inv
`  D )
1710ad2antrr 723 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( F I G ) )  /\  x  e.  B )  ->  D  e.  Cat )
18 fuciso.b . . . . . . . 8  |-  B  =  ( Base `  C
)
19 relfunc 15353 . . . . . . . . 9  |-  Rel  ( C  Func  D )
20 1st2ndbr 6822 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
2119, 6, 20sylancr 661 . . . . . . . 8  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
2218, 15, 21funcf1 15357 . . . . . . 7  |-  ( ph  ->  ( 1st `  F
) : B --> ( Base `  D ) )
2322adantr 463 . . . . . 6  |-  ( (
ph  /\  A  e.  ( F I G ) )  ->  ( 1st `  F ) : B --> ( Base `  D )
)
2423ffvelrnda 6007 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( F I G ) )  /\  x  e.  B )  ->  (
( 1st `  F
) `  x )  e.  ( Base `  D
) )
25 1st2ndbr 6822 . . . . . . . . 9  |-  ( ( Rel  ( C  Func  D )  /\  G  e.  ( C  Func  D
) )  ->  ( 1st `  G ) ( C  Func  D )
( 2nd `  G
) )
2619, 12, 25sylancr 661 . . . . . . . 8  |-  ( ph  ->  ( 1st `  G
) ( C  Func  D ) ( 2nd `  G
) )
2718, 15, 26funcf1 15357 . . . . . . 7  |-  ( ph  ->  ( 1st `  G
) : B --> ( Base `  D ) )
2827adantr 463 . . . . . 6  |-  ( (
ph  /\  A  e.  ( F I G ) )  ->  ( 1st `  G ) : B --> ( Base `  D )
)
2928ffvelrnda 6007 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( F I G ) )  /\  x  e.  B )  ->  (
( 1st `  G
) `  x )  e.  ( Base `  D
) )
30 fuciso.j . . . . 5  |-  J  =  (  Iso  `  D
)
31 eqid 2454 . . . . . . . . . . . 12  |-  (Inv `  Q )  =  (Inv
`  Q )
322, 31, 11, 6, 12, 5isoval 15256 . . . . . . . . . . 11  |-  ( ph  ->  ( F I G )  =  dom  ( F (Inv `  Q ) G ) )
3332eleq2d 2524 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  ( F I G )  <-> 
A  e.  dom  ( F (Inv `  Q ) G ) ) )
342, 31, 11, 6, 12invfun 15255 . . . . . . . . . . 11  |-  ( ph  ->  Fun  ( F (Inv
`  Q ) G ) )
35 funfvbrb 5976 . . . . . . . . . . 11  |-  ( Fun  ( F (Inv `  Q ) G )  ->  ( A  e. 
dom  ( F (Inv
`  Q ) G )  <->  A ( F (Inv
`  Q ) G ) ( ( F (Inv `  Q ) G ) `  A
) ) )
3634, 35syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( A  e.  dom  ( F (Inv `  Q
) G )  <->  A ( F (Inv `  Q ) G ) ( ( F (Inv `  Q
) G ) `  A ) ) )
3733, 36bitrd 253 . . . . . . . . 9  |-  ( ph  ->  ( A  e.  ( F I G )  <-> 
A ( F (Inv
`  Q ) G ) ( ( F (Inv `  Q ) G ) `  A
) ) )
3837biimpa 482 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( F I G ) )  ->  A ( F (Inv `  Q ) G ) ( ( F (Inv `  Q
) G ) `  A ) )
391, 18, 3, 6, 12, 31, 16fucinv 15464 . . . . . . . . 9  |-  ( ph  ->  ( A ( F (Inv `  Q ) G ) ( ( F (Inv `  Q
) G ) `  A )  <->  ( A  e.  ( F N G )  /\  ( ( F (Inv `  Q
) G ) `  A )  e.  ( G N F )  /\  A. x  e.  B  ( A `  x ) ( ( ( 1st `  F
) `  x )
(Inv `  D )
( ( 1st `  G
) `  x )
) ( ( ( F (Inv `  Q
) G ) `  A ) `  x
) ) ) )
4039adantr 463 . . . . . . . 8  |-  ( (
ph  /\  A  e.  ( F I G ) )  ->  ( A
( F (Inv `  Q ) G ) ( ( F (Inv
`  Q ) G ) `  A )  <-> 
( A  e.  ( F N G )  /\  ( ( F (Inv `  Q ) G ) `  A
)  e.  ( G N F )  /\  A. x  e.  B  ( A `  x ) ( ( ( 1st `  F ) `  x
) (Inv `  D
) ( ( 1st `  G ) `  x
) ) ( ( ( F (Inv `  Q ) G ) `
 A ) `  x ) ) ) )
4138, 40mpbid 210 . . . . . . 7  |-  ( (
ph  /\  A  e.  ( F I G ) )  ->  ( A  e.  ( F N G )  /\  ( ( F (Inv `  Q
) G ) `  A )  e.  ( G N F )  /\  A. x  e.  B  ( A `  x ) ( ( ( 1st `  F
) `  x )
(Inv `  D )
( ( 1st `  G
) `  x )
) ( ( ( F (Inv `  Q
) G ) `  A ) `  x
) ) )
4241simp3d 1008 . . . . . 6  |-  ( (
ph  /\  A  e.  ( F I G ) )  ->  A. x  e.  B  ( A `  x ) ( ( ( 1st `  F
) `  x )
(Inv `  D )
( ( 1st `  G
) `  x )
) ( ( ( F (Inv `  Q
) G ) `  A ) `  x
) )
4342r19.21bi 2823 . . . . 5  |-  ( ( ( ph  /\  A  e.  ( F I G ) )  /\  x  e.  B )  ->  ( A `  x )
( ( ( 1st `  F ) `  x
) (Inv `  D
) ( ( 1st `  G ) `  x
) ) ( ( ( F (Inv `  Q ) G ) `
 A ) `  x ) )
4415, 16, 17, 24, 29, 30, 43inviso1 15257 . . . 4  |-  ( ( ( ph  /\  A  e.  ( F I G ) )  /\  x  e.  B )  ->  ( A `  x )  e.  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) )
4544ralrimiva 2868 . . 3  |-  ( (
ph  /\  A  e.  ( F I G ) )  ->  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) )
4614, 45jca 530 . 2  |-  ( (
ph  /\  A  e.  ( F I G ) )  ->  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) )
4711adantr 463 . . 3  |-  ( (
ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) )  ->  Q  e.  Cat )
486adantr 463 . . 3  |-  ( (
ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) )  ->  F  e.  ( C  Func  D )
)
4912adantr 463 . . 3  |-  ( (
ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) )  ->  G  e.  ( C  Func  D )
)
50 simprl 754 . . . 4  |-  ( (
ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) )  ->  A  e.  ( F N G ) )
51 simprr 755 . . . . . . 7  |-  ( (
ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) )  ->  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `
 x ) J ( ( 1st `  G
) `  x )
) )
52 fveq2 5848 . . . . . . . . 9  |-  ( x  =  y  ->  ( A `  x )  =  ( A `  y ) )
53 fveq2 5848 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( 1st `  F
) `  x )  =  ( ( 1st `  F ) `  y
) )
54 fveq2 5848 . . . . . . . . . 10  |-  ( x  =  y  ->  (
( 1st `  G
) `  x )  =  ( ( 1st `  G ) `  y
) )
5553, 54oveq12d 6288 . . . . . . . . 9  |-  ( x  =  y  ->  (
( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  =  ( ( ( 1st `  F
) `  y ) J ( ( 1st `  G ) `  y
) ) )
5652, 55eleq12d 2536 . . . . . . . 8  |-  ( x  =  y  ->  (
( A `  x
)  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) )  <->  ( A `  y )  e.  ( ( ( 1st `  F
) `  y ) J ( ( 1st `  G ) `  y
) ) ) )
5756rspccva 3206 . . . . . . 7  |-  ( ( A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `
 x ) J ( ( 1st `  G
) `  x )
)  /\  y  e.  B )  ->  ( A `  y )  e.  ( ( ( 1st `  F ) `  y
) J ( ( 1st `  G ) `
 y ) ) )
5851, 57sylan 469 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ) )  /\  y  e.  B )  ->  ( A `  y )  e.  ( ( ( 1st `  F ) `  y
) J ( ( 1st `  G ) `
 y ) ) )
5910ad2antrr 723 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ) )  /\  y  e.  B )  ->  D  e.  Cat )
6022adantr 463 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) )  ->  ( 1st `  F
) : B --> ( Base `  D ) )
6160ffvelrnda 6007 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ) )  /\  y  e.  B )  ->  (
( 1st `  F
) `  y )  e.  ( Base `  D
) )
6227adantr 463 . . . . . . . 8  |-  ( (
ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) )  ->  ( 1st `  G
) : B --> ( Base `  D ) )
6362ffvelrnda 6007 . . . . . . 7  |-  ( ( ( ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ) )  /\  y  e.  B )  ->  (
( 1st `  G
) `  y )  e.  ( Base `  D
) )
6415, 16, 59, 61, 63, 30isoval 15256 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ) )  /\  y  e.  B )  ->  (
( ( 1st `  F
) `  y ) J ( ( 1st `  G ) `  y
) )  =  dom  ( ( ( 1st `  F ) `  y
) (Inv `  D
) ( ( 1st `  G ) `  y
) ) )
6558, 64eleqtrd 2544 . . . . 5  |-  ( ( ( ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ) )  /\  y  e.  B )  ->  ( A `  y )  e.  dom  ( ( ( 1st `  F ) `
 y ) (Inv
`  D ) ( ( 1st `  G
) `  y )
) )
6615, 16, 59, 61, 63invfun 15255 . . . . . 6  |-  ( ( ( ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ) )  /\  y  e.  B )  ->  Fun  ( ( ( 1st `  F ) `  y
) (Inv `  D
) ( ( 1st `  G ) `  y
) ) )
67 funfvbrb 5976 . . . . . 6  |-  ( Fun  ( ( ( 1st `  F ) `  y
) (Inv `  D
) ( ( 1st `  G ) `  y
) )  ->  (
( A `  y
)  e.  dom  (
( ( 1st `  F
) `  y )
(Inv `  D )
( ( 1st `  G
) `  y )
)  <->  ( A `  y ) ( ( ( 1st `  F
) `  y )
(Inv `  D )
( ( 1st `  G
) `  y )
) ( ( ( ( 1st `  F
) `  y )
(Inv `  D )
( ( 1st `  G
) `  y )
) `  ( A `  y ) ) ) )
6866, 67syl 16 . . . . 5  |-  ( ( ( ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ) )  /\  y  e.  B )  ->  (
( A `  y
)  e.  dom  (
( ( 1st `  F
) `  y )
(Inv `  D )
( ( 1st `  G
) `  y )
)  <->  ( A `  y ) ( ( ( 1st `  F
) `  y )
(Inv `  D )
( ( 1st `  G
) `  y )
) ( ( ( ( 1st `  F
) `  y )
(Inv `  D )
( ( 1st `  G
) `  y )
) `  ( A `  y ) ) ) )
6965, 68mpbid 210 . . . 4  |-  ( ( ( ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F ) `  x
) J ( ( 1st `  G ) `
 x ) ) ) )  /\  y  e.  B )  ->  ( A `  y )
( ( ( 1st `  F ) `  y
) (Inv `  D
) ( ( 1st `  G ) `  y
) ) ( ( ( ( 1st `  F
) `  y )
(Inv `  D )
( ( 1st `  G
) `  y )
) `  ( A `  y ) ) )
701, 18, 3, 48, 49, 31, 16, 50, 69invfuc 15465 . . 3  |-  ( (
ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) )  ->  A ( F (Inv `  Q ) G ) ( y  e.  B  |->  ( ( ( ( 1st `  F
) `  y )
(Inv `  D )
( ( 1st `  G
) `  y )
) `  ( A `  y ) ) ) )
712, 31, 47, 48, 49, 5, 70inviso1 15257 . 2  |-  ( (
ph  /\  ( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) )  ->  A  e.  ( F I G ) )
7246, 71impbida 830 1  |-  ( ph  ->  ( A  e.  ( F I G )  <-> 
( A  e.  ( F N G )  /\  A. x  e.  B  ( A `  x )  e.  ( ( ( 1st `  F
) `  x ) J ( ( 1st `  G ) `  x
) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   A.wral 2804   class class class wbr 4439    |-> cmpt 4497   dom cdm 4988   Rel wrel 4993   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270   1stc1st 6771   2ndc2nd 6772   Basecbs 14719   Catccat 15156  Invcinv 15236    Iso ciso 15237    Func cfunc 15345   Nat cnat 15432   FuncCat cfuc 15433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-fz 11676  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-hom 14811  df-cco 14812  df-cat 15160  df-cid 15161  df-sect 15238  df-inv 15239  df-iso 15240  df-func 15349  df-nat 15434  df-fuc 15435
This theorem is referenced by:  yonffthlem  15753
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