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Theorem estrres 16012
Description: Any restriction of a category (as an extensible structure which is an unordered triple of ordered pairs) is an unordered triple of ordered pairs. (Contributed by AV, 15-Mar-2020.)
Hypotheses
Ref Expression
estrres.c  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
estrres.b  |-  ( ph  ->  B  e.  V )
estrres.h  |-  ( ph  ->  H  e.  X )
estrres.x  |-  ( ph  ->  .x.  e.  Y )
estrres.a  |-  ( ph  ->  A  e.  U )
estrres.g  |-  ( ph  ->  G  e.  W )
estrres.u  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
estrres  |-  ( ph  ->  ( ( Cs  A ) sSet  <. ( Hom  `  ndx ) ,  G >. )  =  { <. ( Base `  ndx ) ,  A >. ,  <. ( Hom  `  ndx ) ,  G >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )

Proof of Theorem estrres
StepHypRef Expression
1 ovex 6330 . . 3  |-  ( Cs  A )  e.  _V
2 estrres.g . . 3  |-  ( ph  ->  G  e.  W )
3 setsval 15134 . . 3  |-  ( ( ( Cs  A )  e.  _V  /\  G  e.  W )  ->  ( ( Cs  A ) sSet  <. ( Hom  `  ndx ) ,  G >. )  =  ( ( ( Cs  A )  |`  ( _V  \  { ( Hom  `  ndx ) } ) )  u.  { <. ( Hom  `  ndx ) ,  G >. } ) )
41, 2, 3sylancr 667 . 2  |-  ( ph  ->  ( ( Cs  A ) sSet  <. ( Hom  `  ndx ) ,  G >. )  =  ( ( ( Cs  A )  |`  ( _V  \  { ( Hom  `  ndx ) } ) )  u.  { <. ( Hom  `  ndx ) ,  G >. } ) )
5 eqid 2422 . . . . . 6  |-  ( Cs  A )  =  ( Cs  A )
6 eqid 2422 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
7 eqid 2422 . . . . . 6  |-  ( Base `  ndx )  =  (
Base `  ndx )
8 estrres.c . . . . . . 7  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
9 tpex 6601 . . . . . . 7  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. }  e.  _V
108, 9syl6eqel 2518 . . . . . 6  |-  ( ph  ->  C  e.  _V )
11 fvex 5888 . . . . . . . . . 10  |-  ( Base `  ndx )  e.  _V
12 fvex 5888 . . . . . . . . . 10  |-  ( Hom  `  ndx )  e.  _V
13 fvex 5888 . . . . . . . . . 10  |-  (comp `  ndx )  e.  _V
1411, 12, 133pm3.2i 1183 . . . . . . . . 9  |-  ( (
Base `  ndx )  e. 
_V  /\  ( Hom  ` 
ndx )  e.  _V  /\  (comp `  ndx )  e. 
_V )
1514a1i 11 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  ndx )  e.  _V  /\  ( Hom  `  ndx )  e. 
_V  /\  (comp `  ndx )  e.  _V )
)
16 estrres.b . . . . . . . 8  |-  ( ph  ->  B  e.  V )
17 estrres.h . . . . . . . 8  |-  ( ph  ->  H  e.  X )
18 estrres.x . . . . . . . 8  |-  ( ph  ->  .x.  e.  Y )
19 slotsbhcdif 15306 . . . . . . . . 9  |-  ( (
Base `  ndx )  =/=  ( Hom  `  ndx )  /\  ( Base `  ndx )  =/=  (comp `  ndx )  /\  ( Hom  `  ndx )  =/=  (comp `  ndx ) )
2019a1i 11 . . . . . . . 8  |-  ( ph  ->  ( ( Base `  ndx )  =/=  ( Hom  `  ndx )  /\  ( Base `  ndx )  =/=  (comp `  ndx )  /\  ( Hom  `  ndx )  =/=  (comp `  ndx ) ) )
21 funtpg 5648 . . . . . . . 8  |-  ( ( ( ( Base `  ndx )  e.  _V  /\  ( Hom  `  ndx )  e. 
_V  /\  (comp `  ndx )  e.  _V )  /\  ( B  e.  V  /\  H  e.  X  /\  .x.  e.  Y )  /\  ( ( Base `  ndx )  =/=  ( Hom  `  ndx )  /\  ( Base `  ndx )  =/=  (comp `  ndx )  /\  ( Hom  `  ndx )  =/=  (comp `  ndx ) ) )  ->  Fun  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
2215, 16, 17, 18, 20, 21syl131anc 1277 . . . . . . 7  |-  ( ph  ->  Fun  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
238funeqd 5619 . . . . . . 7  |-  ( ph  ->  ( Fun  C  <->  Fun  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } ) )
2422, 23mpbird 235 . . . . . 6  |-  ( ph  ->  Fun  C )
258, 16, 17, 18estrreslem2 16011 . . . . . 6  |-  ( ph  ->  ( Base `  ndx )  e.  dom  C )
26 estrres.a . . . . . 6  |-  ( ph  ->  A  e.  U )
27 estrres.u . . . . . . 7  |-  ( ph  ->  A  C_  B )
288, 16estrreslem1 16010 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  C ) )
2927, 28sseqtrd 3500 . . . . . 6  |-  ( ph  ->  A  C_  ( Base `  C ) )
305, 6, 7, 10, 24, 25, 26, 29ressval3d 15174 . . . . 5  |-  ( ph  ->  ( Cs  A )  =  ( C sSet  <. ( Base `  ndx ) ,  A >. ) )
3130reseq1d 5120 . . . 4  |-  ( ph  ->  ( ( Cs  A )  |`  ( _V  \  {
( Hom  `  ndx ) } ) )  =  ( ( C sSet  <. (
Base `  ndx ) ,  A >. )  |`  ( _V  \  { ( Hom  `  ndx ) } ) ) )
3231uneq1d 3619 . . 3  |-  ( ph  ->  ( ( ( Cs  A )  |`  ( _V  \  { ( Hom  `  ndx ) } ) )  u. 
{ <. ( Hom  `  ndx ) ,  G >. } )  =  ( ( ( C sSet  <. ( Base `  ndx ) ,  A >. )  |`  ( _V  \  { ( Hom  `  ndx ) } ) )  u.  { <. ( Hom  `  ndx ) ,  G >. } ) )
33 setsval 15134 . . . . . . . 8  |-  ( ( C  e.  _V  /\  A  e.  U )  ->  ( C sSet  <. ( Base `  ndx ) ,  A >. )  =  ( ( C  |`  ( _V  \  { ( Base `  ndx ) } ) )  u.  { <. (
Base `  ndx ) ,  A >. } ) )
3410, 26, 33syl2anc 665 . . . . . . 7  |-  ( ph  ->  ( C sSet  <. ( Base `  ndx ) ,  A >. )  =  ( ( C  |`  ( _V  \  { ( Base `  ndx ) } ) )  u.  { <. (
Base `  ndx ) ,  A >. } ) )
3534reseq1d 5120 . . . . . 6  |-  ( ph  ->  ( ( C sSet  <. (
Base `  ndx ) ,  A >. )  |`  ( _V  \  { ( Hom  `  ndx ) } ) )  =  ( ( ( C  |`  ( _V  \  { ( Base `  ndx ) } ) )  u.  { <. (
Base `  ndx ) ,  A >. } )  |`  ( _V  \  { ( Hom  `  ndx ) } ) ) )
3612a1i 11 . . . . . . . . . 10  |-  ( ph  ->  ( Hom  `  ndx )  e.  _V )
3713a1i 11 . . . . . . . . . 10  |-  ( ph  ->  (comp `  ndx )  e. 
_V )
38 elex 3090 . . . . . . . . . . 11  |-  ( H  e.  X  ->  H  e.  _V )
3917, 38syl 17 . . . . . . . . . 10  |-  ( ph  ->  H  e.  _V )
40 elex 3090 . . . . . . . . . . 11  |-  (  .x.  e.  Y  ->  .x.  e.  _V )
4118, 40syl 17 . . . . . . . . . 10  |-  ( ph  ->  .x.  e.  _V )
42 simp1 1005 . . . . . . . . . . . 12  |-  ( ( ( Base `  ndx )  =/=  ( Hom  `  ndx )  /\  ( Base `  ndx )  =/=  (comp `  ndx )  /\  ( Hom  `  ndx )  =/=  (comp `  ndx ) )  ->  ( Base `  ndx )  =/=  ( Hom  `  ndx ) )
4342necomd 2695 . . . . . . . . . . 11  |-  ( ( ( Base `  ndx )  =/=  ( Hom  `  ndx )  /\  ( Base `  ndx )  =/=  (comp `  ndx )  /\  ( Hom  `  ndx )  =/=  (comp `  ndx ) )  ->  ( Hom  `  ndx )  =/=  ( Base `  ndx ) )
4419, 43mp1i 13 . . . . . . . . . 10  |-  ( ph  ->  ( Hom  `  ndx )  =/=  ( Base `  ndx ) )
45 simp2 1006 . . . . . . . . . . . 12  |-  ( ( ( Base `  ndx )  =/=  ( Hom  `  ndx )  /\  ( Base `  ndx )  =/=  (comp `  ndx )  /\  ( Hom  `  ndx )  =/=  (comp `  ndx ) )  ->  ( Base `  ndx )  =/=  (comp `  ndx ) )
4645necomd 2695 . . . . . . . . . . 11  |-  ( ( ( Base `  ndx )  =/=  ( Hom  `  ndx )  /\  ( Base `  ndx )  =/=  (comp `  ndx )  /\  ( Hom  `  ndx )  =/=  (comp `  ndx ) )  ->  (comp ` 
ndx )  =/=  ( Base `  ndx ) )
4719, 46mp1i 13 . . . . . . . . . 10  |-  ( ph  ->  (comp `  ndx )  =/=  ( Base `  ndx ) )
488, 36, 37, 39, 41, 44, 47tpres 6129 . . . . . . . . 9  |-  ( ph  ->  ( C  |`  ( _V  \  { ( Base `  ndx ) } ) )  =  { <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
4948uneq1d 3619 . . . . . . . 8  |-  ( ph  ->  ( ( C  |`  ( _V  \  { (
Base `  ndx ) } ) )  u.  { <. ( Base `  ndx ) ,  A >. } )  =  ( {
<. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. }  u.  { <. ( Base `  ndx ) ,  A >. } ) )
50 df-tp 4001 . . . . . . . 8  |-  { <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. ,  <. ( Base `  ndx ) ,  A >. }  =  ( { <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. }  u.  { <. ( Base `  ndx ) ,  A >. } )
5149, 50syl6eqr 2481 . . . . . . 7  |-  ( ph  ->  ( ( C  |`  ( _V  \  { (
Base `  ndx ) } ) )  u.  { <. ( Base `  ndx ) ,  A >. } )  =  { <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. ,  <. ( Base `  ndx ) ,  A >. } )
5211a1i 11 . . . . . . 7  |-  ( ph  ->  ( Base `  ndx )  e.  _V )
5316, 27ssexd 4568 . . . . . . 7  |-  ( ph  ->  A  e.  _V )
54 simp3 1007 . . . . . . . . 9  |-  ( ( ( Base `  ndx )  =/=  ( Hom  `  ndx )  /\  ( Base `  ndx )  =/=  (comp `  ndx )  /\  ( Hom  `  ndx )  =/=  (comp `  ndx ) )  ->  ( Hom  `  ndx )  =/=  (comp `  ndx ) )
5554necomd 2695 . . . . . . . 8  |-  ( ( ( Base `  ndx )  =/=  ( Hom  `  ndx )  /\  ( Base `  ndx )  =/=  (comp `  ndx )  /\  ( Hom  `  ndx )  =/=  (comp `  ndx ) )  ->  (comp ` 
ndx )  =/=  ( Hom  `  ndx ) )
5619, 55mp1i 13 . . . . . . 7  |-  ( ph  ->  (comp `  ndx )  =/=  ( Hom  `  ndx ) )
5719, 42mp1i 13 . . . . . . 7  |-  ( ph  ->  ( Base `  ndx )  =/=  ( Hom  `  ndx ) )
5851, 37, 52, 41, 53, 56, 57tpres 6129 . . . . . 6  |-  ( ph  ->  ( ( ( C  |`  ( _V  \  {
( Base `  ndx ) } ) )  u.  { <. ( Base `  ndx ) ,  A >. } )  |`  ( _V  \  { ( Hom  `  ndx ) } ) )  =  { <. (comp `  ndx ) ,  .x.  >. ,  <. (
Base `  ndx ) ,  A >. } )
5935, 58eqtrd 2463 . . . . 5  |-  ( ph  ->  ( ( C sSet  <. (
Base `  ndx ) ,  A >. )  |`  ( _V  \  { ( Hom  `  ndx ) } ) )  =  { <. (comp `  ndx ) ,  .x.  >. ,  <. ( Base `  ndx ) ,  A >. } )
6059uneq1d 3619 . . . 4  |-  ( ph  ->  ( ( ( C sSet  <. ( Base `  ndx ) ,  A >. )  |`  ( _V  \  {
( Hom  `  ndx ) } ) )  u. 
{ <. ( Hom  `  ndx ) ,  G >. } )  =  ( {
<. (comp `  ndx ) , 
.x.  >. ,  <. ( Base `  ndx ) ,  A >. }  u.  { <. ( Hom  `  ndx ) ,  G >. } ) )
61 df-tp 4001 . . . . . 6  |-  { <. (comp `  ndx ) ,  .x.  >. ,  <. ( Base `  ndx ) ,  A >. , 
<. ( Hom  `  ndx ) ,  G >. }  =  ( { <. (comp `  ndx ) ,  .x.  >. ,  <. ( Base `  ndx ) ,  A >. }  u.  { <. ( Hom  `  ndx ) ,  G >. } )
62 tprot 4092 . . . . . 6  |-  { <. (comp `  ndx ) ,  .x.  >. ,  <. ( Base `  ndx ) ,  A >. , 
<. ( Hom  `  ndx ) ,  G >. }  =  { <. ( Base `  ndx ) ,  A >. ,  <. ( Hom  `  ndx ) ,  G >. ,  <. (comp ` 
ndx ) ,  .x.  >. }
6361, 62eqtr3i 2453 . . . . 5  |-  ( {
<. (comp `  ndx ) , 
.x.  >. ,  <. ( Base `  ndx ) ,  A >. }  u.  { <. ( Hom  `  ndx ) ,  G >. } )  =  { <. (
Base `  ndx ) ,  A >. ,  <. ( Hom  `  ndx ) ,  G >. ,  <. (comp ` 
ndx ) ,  .x.  >. }
6463a1i 11 . . . 4  |-  ( ph  ->  ( { <. (comp ` 
ndx ) ,  .x.  >. ,  <. ( Base `  ndx ) ,  A >. }  u.  { <. ( Hom  `  ndx ) ,  G >. } )  =  { <. ( Base `  ndx ) ,  A >. , 
<. ( Hom  `  ndx ) ,  G >. , 
<. (comp `  ndx ) , 
.x.  >. } )
6560, 64eqtrd 2463 . . 3  |-  ( ph  ->  ( ( ( C sSet  <. ( Base `  ndx ) ,  A >. )  |`  ( _V  \  {
( Hom  `  ndx ) } ) )  u. 
{ <. ( Hom  `  ndx ) ,  G >. } )  =  { <. (
Base `  ndx ) ,  A >. ,  <. ( Hom  `  ndx ) ,  G >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
6632, 65eqtrd 2463 . 2  |-  ( ph  ->  ( ( ( Cs  A )  |`  ( _V  \  { ( Hom  `  ndx ) } ) )  u. 
{ <. ( Hom  `  ndx ) ,  G >. } )  =  { <. (
Base `  ndx ) ,  A >. ,  <. ( Hom  `  ndx ) ,  G >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
674, 66eqtrd 2463 1  |-  ( ph  ->  ( ( Cs  A ) sSet  <. ( Hom  `  ndx ) ,  G >. )  =  { <. ( Base `  ndx ) ,  A >. ,  <. ( Hom  `  ndx ) ,  G >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1868    =/= wne 2618   _Vcvv 3081    \ cdif 3433    u. cun 3434    C_ wss 3436   {csn 3996   {cpr 3998   {ctp 4000   <.cop 4002    |` cres 4852   Fun wfun 5592   ` cfv 5598  (class class class)co 6302   ndxcnx 15106   sSet csts 15107   Basecbs 15109   ↾s cress 15110   Hom chom 15189  compcco 15190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-om 6704  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-2 10669  df-3 10670  df-4 10671  df-5 10672  df-6 10673  df-7 10674  df-8 10675  df-9 10676  df-10 10677  df-n0 10871  df-z 10939  df-dec 11053  df-ndx 15112  df-slot 15113  df-base 15114  df-sets 15115  df-ress 15116  df-hom 15202  df-cco 15203
This theorem is referenced by:  dfrngc2  39246  dfringc2  39292  rngcresringcat  39304
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