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Theorem pwssplit1 17152
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 17151 . 2  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B --> C )
7 simp1 988 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  W  e.  Mnd )
8 simp2 989 . . . . . . . . . 10  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  U  e.  X )
9 simp3 990 . . . . . . . . . 10  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  V  C_  U )
108, 9ssexd 4451 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  V  e.  _V )
11 eqid 2443 . . . . . . . . . 10  |-  ( Base `  W )  =  (
Base `  W )
122, 11, 4pwselbasb 14438 . . . . . . . . 9  |-  ( ( W  e.  Mnd  /\  V  e.  _V )  ->  ( a  e.  C  <->  a : V --> ( Base `  W ) ) )
137, 10, 12syl2anc 661 . . . . . . . 8  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  -> 
( a  e.  C  <->  a : V --> ( Base `  W ) ) )
1413biimpa 484 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  a : V --> ( Base `  W ) )
15 fvex 5713 . . . . . . . . . 10  |-  ( 0g
`  W )  e. 
_V
1615fconst 5608 . . . . . . . . 9  |-  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) : ( U 
\  V ) --> { ( 0g `  W
) }
1716a1i 11 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> { ( 0g
`  W ) } )
18 simpl1 991 . . . . . . . . . 10  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  W  e.  Mnd )
19 eqid 2443 . . . . . . . . . . 11  |-  ( 0g
`  W )  =  ( 0g `  W
)
2011, 19mndidcl 15451 . . . . . . . . . 10  |-  ( W  e.  Mnd  ->  ( 0g `  W )  e.  ( Base `  W
) )
2118, 20syl 16 . . . . . . . . 9  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( 0g `  W
)  e.  ( Base `  W ) )
2221snssd 4030 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  { ( 0g `  W ) }  C_  ( Base `  W )
)
23 fss 5579 . . . . . . . 8  |-  ( ( ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> { ( 0g
`  W ) }  /\  { ( 0g
`  W ) } 
C_  ( Base `  W
) )  ->  (
( U  \  V
)  X.  { ( 0g `  W ) } ) : ( U  \  V ) --> ( Base `  W
) )
2417, 22, 23syl2anc 661 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( U  \  V )  X.  {
( 0g `  W
) } ) : ( U  \  V
) --> ( Base `  W
) )
25 disjdif 3763 . . . . . . . 8  |-  ( V  i^i  ( U  \  V ) )  =  (/)
2625a1i 11 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( V  i^i  ( U  \  V ) )  =  (/) )
27 fun 5587 . . . . . . 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 ) ) )
2814, 24, 26, 27syl21anc 1217 . . . . . 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 ) ) )
29 simpl3 993 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  V  C_  U )
30 undif 3771 . . . . . . . 8  |-  ( V 
C_  U  <->  ( V  u.  ( U  \  V
) )  =  U )
3129, 30sylib 196 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( V  u.  ( U  \  V ) )  =  U )
32 unidm 3511 . . . . . . . 8  |-  ( (
Base `  W )  u.  ( Base `  W
) )  =  (
Base `  W )
3332a1i 11 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( Base `  W
)  u.  ( Base `  W ) )  =  ( Base `  W
) )
3431, 33feq23d 5566 . . . . . 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
) ) )
3528, 34mpbid 210 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) ) : U --> ( Base `  W
) )
36 simpl2 992 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  U  e.  X )
371, 11, 3pwselbasb 14438 . . . . . 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 ) ) )
3818, 36, 37syl2anc 661 . . . . 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 ) ) )
3935, 38mpbird 232 . . . 4  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  u.  (
( U  \  V
)  X.  { ( 0g `  W ) } ) )  e.  B )
405fvtresfn 5787 . . . . . 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 ) )
4139, 40syl 16 . . . . 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 ) )
42 resundir 5137 . . . . . . 7  |-  ( ( a  u.  ( ( U  \  V )  X.  { ( 0g
`  W ) } ) )  |`  V )  =  ( ( a  |`  V )  u.  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V ) )
43 ffn 5571 . . . . . . . . 9  |-  ( a : V --> ( Base `  W )  ->  a  Fn  V )
44 fnresdm 5532 . . . . . . . . 9  |-  ( a  Fn  V  ->  (
a  |`  V )  =  a )
4514, 43, 443syl 20 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( a  |`  V )  =  a )
46 incom 3555 . . . . . . . . . 10  |-  ( ( U  \  V )  i^i  V )  =  ( V  i^i  ( U  \  V ) )
4746, 25eqtri 2463 . . . . . . . . 9  |-  ( ( U  \  V )  i^i  V )  =  (/)
48 fnconstg 5610 . . . . . . . . . . 11  |-  ( ( 0g `  W )  e.  _V  ->  (
( U  \  V
)  X.  { ( 0g `  W ) } )  Fn  ( U  \  V ) )
4915, 48ax-mp 5 . . . . . . . . . 10  |-  ( ( U  \  V )  X.  { ( 0g
`  W ) } )  Fn  ( U 
\  V )
50 fnresdisj 5533 . . . . . . . . . 10  |-  ( ( ( U  \  V
)  X.  { ( 0g `  W ) } )  Fn  ( U  \  V )  -> 
( ( ( U 
\  V )  i^i 
V )  =  (/)  <->  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V )  =  (/) ) )
5149, 50mp1i 12 . . . . . . . . 9  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( ( U 
\  V )  i^i 
V )  =  (/)  <->  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V )  =  (/) ) )
5247, 51mpbii 211 . . . . . . . 8  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( ( U 
\  V )  X. 
{ ( 0g `  W ) } )  |`  V )  =  (/) )
5345, 52uneq12d 3523 . . . . . . 7  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  |`  V )  u.  (
( ( U  \  V )  X.  {
( 0g `  W
) } )  |`  V ) )  =  ( a  u.  (/) ) )
5442, 53syl5eq 2487 . . . . . 6  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V )  =  ( a  u.  (/) ) )
55 un0 3674 . . . . . 6  |-  ( a  u.  (/) )  =  a
5654, 55syl6eq 2491 . . . . 5  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  ( ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  |`  V )  =  a )
5741, 56eqtr2d 2476 . . . 4  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  a  =  ( F `
 ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) ) ) )
58 fveq2 5703 . . . . . 6  |-  ( b  =  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  ->  ( F `  b )  =  ( F `  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) ) )
5958eqeq2d 2454 . . . . 5  |-  ( b  =  ( a  u.  ( ( U  \  V )  X.  {
( 0g `  W
) } ) )  ->  ( a  =  ( F `  b
)  <->  a  =  ( F `  ( a  u.  ( ( U 
\  V )  X. 
{ ( 0g `  W ) } ) ) ) ) )
6059rspcev 3085 . . . 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 ) )
6139, 57, 60syl2anc 661 . . 3  |-  ( ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  /\  a  e.  C )  ->  E. b  e.  B  a  =  ( F `  b ) )
6261ralrimiva 2811 . 2  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  A. a  e.  C  E. b  e.  B  a  =  ( F `  b ) )
63 dffo3 5870 . 2  |-  ( F : B -onto-> C  <->  ( F : B --> C  /\  A. a  e.  C  E. b  e.  B  a  =  ( F `  b ) ) )
646, 62, 63sylanbrc 664 1  |-  ( ( W  e.  Mnd  /\  U  e.  X  /\  V  C_  U )  ->  F : B -onto-> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   E.wrex 2728   _Vcvv 2984    \ cdif 3337    u. cun 3338    i^i cin 3339    C_ wss 3340   (/)c0 3649   {csn 3889    e. cmpt 4362    X. cxp 4850    |` cres 4854    Fn wfn 5425   -->wf 5426   -onto->wfo 5428   ` cfv 5430  (class class class)co 6103   Basecbs 14186   0gc0g 14390    ^s cpws 14397   Mndcmnd 15421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-ixp 7276  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-4 10394  df-5 10395  df-6 10396  df-7 10397  df-8 10398  df-9 10399  df-10 10400  df-n0 10592  df-z 10659  df-dec 10768  df-uz 10874  df-fz 11450  df-struct 14188  df-ndx 14189  df-slot 14190  df-base 14191  df-plusg 14263  df-mulr 14264  df-sca 14266  df-vsca 14267  df-ip 14268  df-tset 14269  df-ple 14270  df-ds 14272  df-hom 14274  df-cco 14275  df-0g 14392  df-prds 14398  df-pws 14400  df-mnd 15427
This theorem is referenced by:  pwslnmlem2  29458
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