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Theorem pwssplit1 27291
Description: Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Hypotheses
Ref Expression
pwssplit1.y  |-  Y  =  ( W  ^s  U )
pwssplit1.z  |-  Z  =  ( W  ^s  V )
pwssplit1.b  |-  B  =  ( Base `  Y
)
pwssplit1.c  |-  C  =  ( Base `  Z
)
pwssplit1.f  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
Assertion
Ref Expression
pwssplit1  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B -onto-> C )
Distinct variable groups:    x, Y    x, W    x, U    x, Z    x, V    x, B    x, C    x, X
Allowed substitution hint:    F( x)

Proof of Theorem pwssplit1
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwssplit1.y . . 3  |-  Y  =  ( W  ^s  U )
2 pwssplit1.z . . 3  |-  Z  =  ( W  ^s  V )
3 pwssplit1.b . . 3  |-  B  =  ( Base `  Y
)
4 pwssplit1.c . . 3  |-  C  =  ( Base `  Z
)
5 pwssplit1.f . . 3  |-  F  =  ( x  e.  B  |->  ( x  |`  V ) )
61, 2, 3, 4, 5pwssplit0 27290 . 2  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
7 simp1 955 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  Mnd )
8 simp3 957 . . . . . . . . . 10  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
9 simp2 956 . . . . . . . . . 10  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
10 ssexg 4176 . . . . . . . . . 10  |-  ( ( V  C_  U  /\  U  e.  X )  ->  V  e.  _V )
118, 9, 10syl2anc 642 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
12 eqid 2296 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
132, 12, 4pwselbasb 13403 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  V  e.  _V )  ->  ( a  e.  C  <->  a : V --> ( Base `  W ) ) )
147, 11, 13syl2anc 642 . . . . . . . 8  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  -> 
( a  e.  C  <->  a : V --> ( Base `  W ) ) )
1514biimpa 470 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  a : V --> ( Base `  W ) )
16 fvex 5555 . . . . . . . . . 10  |-  ( 0g
`  W )  e. 
_V
1716fconst 5443 . . . . . . . . 9  |-  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) : ( U 
\  V ) --> { ( 0g `  W
) }
1817a1i 10 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> { ( 0g
`  W ) } )
19 simpl1 958 . . . . . . . . . 10  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  W  e.  Mnd )
20 eqid 2296 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
2112, 20mndidcl 14407 . . . . . . . . . 10  |-  ( W  e.  Mnd  ->  ( 0g `  W )  e.  ( Base `  W
) )
2219, 21syl 15 . . . . . . . . 9  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( 0g `  W
)  e.  ( Base `  W ) )
2322snssd 3776 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  { ( 0g `  W ) }  C_  ( Base `  W )
)
24 fss 5413 . . . . . . . 8  |-  ( ( ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> { ( 0g
`  W ) }  /\  { ( 0g
`  W ) } 
C_  ( Base `  W
) )  ->  (
( U  \  V
)  X.  { ( 0g `  W ) } ) : ( U  \  V ) --> ( Base `  W
) )
2518, 23, 24syl2anc 642 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> ( Base `  W
) )
26 disjdif 3539 . . . . . . . 8  |-  ( V  i^i  ( U  \  V ) )  =  (/)
2726a1i 10 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( V  i^i  ( U  \  V ) )  =  (/) )
28 fun 5421 . . . . . . 7  |-  ( ( ( a : V --> ( Base `  W )  /\  ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> ( Base `  W
) )  /\  ( V  i^i  ( U  \  V ) )  =  (/) )  ->  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) : ( V  u.  ( U  \  V ) ) --> ( ( Base `  W
)  u.  ( Base `  W ) ) )
2915, 25, 27, 28syl21anc 1181 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) ) : ( V  u.  ( U  \  V ) ) --> ( ( Base `  W
)  u.  ( Base `  W ) ) )
30 simpl3 960 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  V  C_  U )
31 undif 3547 . . . . . . . 8  |-  ( V 
C_  U  <->  ( V  u.  ( U  \  V
) )  =  U )
3230, 31sylib 188 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( V  u.  ( U  \  V ) )  =  U )
33 unidm 3331 . . . . . . . 8  |-  ( (
Base `  W )  u.  ( Base `  W
) )  =  (
Base `  W )
3433a1i 10 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( Base `  W
)  u.  ( Base `  W ) )  =  ( Base `  W
) )
3532, 34feq23d 5402 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) : ( V  u.  ( U  \  V ) ) --> ( ( Base `  W )  u.  ( Base `  W ) )  <-> 
( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) ) : U --> ( Base `  W
) ) )
3629, 35mpbid 201 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) ) : U --> ( Base `  W
) )
37 simpl2 959 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  U  e.  X )
381, 12, 3pwselbasb 13403 . . . . . 6  |-  ( ( W  e.  Mnd  /\  U  e.  X )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  e.  B  <->  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) : U --> ( Base `  W ) ) )
3919, 37, 38syl2anc 642 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  e.  B  <->  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) : U --> ( Base `  W ) ) )
4036, 39mpbird 223 . . . 4  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) )  e.  B )
415fvtresfn 26866 . . . . . 6  |-  ( ( a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) )  e.  B  ->  ( F `  (
a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) ) )  =  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V ) )
4240, 41syl 15 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( F `  (
a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) ) )  =  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V ) )
43 resundir 4986 . . . . . . 7  |-  ( ( a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) )  |`  V )  =  ( ( a  |`  V )  u.  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V ) )
44 ffn 5405 . . . . . . . . 9  |-  ( a : V --> ( Base `  W )  ->  a  Fn  V )
45 fnresdm 5369 . . . . . . . . 9  |-  ( a  Fn  V  ->  (
a  |`  V )  =  a )
4615, 44, 453syl 18 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  |`  V )  =  a )
47 incom 3374 . . . . . . . . . 10  |-  ( ( U  \  V )  i^i  V )  =  ( V  i^i  ( U  \  V ) )
4847, 26eqtri 2316 . . . . . . . . 9  |-  ( ( U  \  V )  i^i  V )  =  (/)
49 fnconstg 5445 . . . . . . . . . . 11  |-  ( ( 0g `  W )  e.  _V  ->  (
( U  \  V
)  X.  { ( 0g `  W ) } )  Fn  ( U  \  V ) )
5016, 49ax-mp 8 . . . . . . . . . 10  |-  ( ( U  \  V )  X.  { ( 0g
`  W ) } )  Fn  ( U 
\  V )
51 fnresdisj 5370 . . . . . . . . . 10  |-  ( ( ( U  \  V
)  X.  { ( 0g `  W ) } )  Fn  ( U  \  V )  -> 
( ( ( U 
\  V )  i^i 
V )  =  (/)  <->  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V )  =  (/) ) )
5250, 51mp1i 11 . . . . . . . . 9  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( ( U 
\  V )  i^i 
V )  =  (/)  <->  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V )  =  (/) ) )
5348, 52mpbii 202 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( ( U 
\  V )  X. 
{ ( 0g `  W ) } )  |`  V )  =  (/) )
5446, 53uneq12d 3343 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  |`  V )  u.  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V ) )  =  ( a  u.  (/) ) )
5543, 54syl5eq 2340 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V )  =  ( a  u.  (/) ) )
56 un0 3492 . . . . . 6  |-  ( a  u.  (/) )  =  a
5755, 56syl6eq 2344 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V )  =  a )
5842, 57eqtr2d 2329 . . . 4  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  a  =  ( F `
 ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) ) )
59 fveq2 5541 . . . . . 6  |-  ( b  =  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  ->  ( F `  b )  =  ( F `  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) ) )
6059eqeq2d 2307 . . . . 5  |-  ( b  =  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  ->  ( a  =  ( F `  b
)  <->  a  =  ( F `  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) ) ) )
6160rspcev 2897 . . . 4  |-  ( ( ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) )  e.  B  /\  a  =  ( F `  (
a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) ) ) )  ->  E. b  e.  B  a  =  ( F `  b ) )
6240, 58, 61syl2anc 642 . . 3  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  E. b  e.  B  a  =  ( F `  b ) )
6362ralrimiva 2639 . 2  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  A. a  e.  C  E. b  e.  B  a  =  ( F `  b ) )
64 dffo3 5691 . 2  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  A. a  e.  C  E. b  e.  B  a  =  ( F `  b ) ) )
656, 63, 64sylanbrc 645 1  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B -onto-> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653    e. cmpt 4093    X. cxp 4703    |` cres 4707    Fn wfn 5266   -->wf 5267   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   Basecbs 13164    ^s cpws 13363   0gc0g 13416   Mndcmnd 14377
This theorem is referenced by:  pwslnmlem2  27298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-prds 13364  df-pws 13366  df-0g 13420  df-mnd 14383
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