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Theorem imasaddflem 13432
Description: The image set operations are closed if the original operation is. (Contributed by Mario Carneiro, 23-Feb-2015.)
Hypotheses
Ref Expression
imasaddf.f  |-  ( ph  ->  F : V -onto-> B
)
imasaddf.e  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( F `  a
)  =  ( F `
 p )  /\  ( F `  b )  =  ( F `  q ) )  -> 
( F `  (
a  .x.  b )
)  =  ( F `
 ( p  .x.  q ) ) ) )
imasaddflem.a  |-  ( ph  -> 
.xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
imasaddflem.c  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  -> 
( p  .x.  q
)  e.  V )
Assertion
Ref Expression
imasaddflem  |-  ( ph  -> 
.xb  : ( B  X.  B ) --> B )
Distinct variable groups:    q, p, B    a, b, p, q, V    .x. , p, q    F, a, b, p, q    ph, a,
b, p, q    .xb , a,
b, p, q
Allowed substitution hints:    B( a, b)    .x. ( a, b)

Proof of Theorem imasaddflem
StepHypRef Expression
1 imasaddf.f . . 3  |-  ( ph  ->  F : V -onto-> B
)
2 imasaddf.e . . 3  |-  ( (
ph  /\  ( a  e.  V  /\  b  e.  V )  /\  (
p  e.  V  /\  q  e.  V )
)  ->  ( (
( F `  a
)  =  ( F `
 p )  /\  ( F `  b )  =  ( F `  q ) )  -> 
( F `  (
a  .x.  b )
)  =  ( F `
 ( p  .x.  q ) ) ) )
3 imasaddflem.a . . 3  |-  ( ph  -> 
.xb  =  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. } )
41, 2, 3imasaddfnlem 13430 . 2  |-  ( ph  -> 
.xb  Fn  ( B  X.  B ) )
5 fof 5451 . . . . . . . . . . . 12  |-  ( F : V -onto-> B  ->  F : V --> B )
61, 5syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F : V --> B )
7 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( F : V --> B  /\  p  e.  V )  ->  ( F `  p
)  e.  B )
8 ffvelrn 5663 . . . . . . . . . . . . 13  |-  ( ( F : V --> B  /\  q  e.  V )  ->  ( F `  q
)  e.  B )
97, 8anim12dan 810 . . . . . . . . . . . 12  |-  ( ( F : V --> B  /\  ( p  e.  V  /\  q  e.  V
) )  ->  (
( F `  p
)  e.  B  /\  ( F `  q )  e.  B ) )
10 opelxpi 4721 . . . . . . . . . . . 12  |-  ( ( ( F `  p
)  e.  B  /\  ( F `  q )  e.  B )  ->  <. ( F `  p
) ,  ( F `
 q ) >.  e.  ( B  X.  B
) )
119, 10syl 15 . . . . . . . . . . 11  |-  ( ( F : V --> B  /\  ( p  e.  V  /\  q  e.  V
) )  ->  <. ( F `  p ) ,  ( F `  q ) >.  e.  ( B  X.  B ) )
126, 11sylan 457 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  <. ( F `  p
) ,  ( F `
 q ) >.  e.  ( B  X.  B
) )
13 imasaddflem.c . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  -> 
( p  .x.  q
)  e.  V )
14 ffvelrn 5663 . . . . . . . . . . . 12  |-  ( ( F : V --> B  /\  ( p  .x.  q )  e.  V )  -> 
( F `  (
p  .x.  q )
)  e.  B )
156, 14sylan 457 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  .x.  q )  e.  V
)  ->  ( F `  ( p  .x.  q
) )  e.  B
)
1613, 15syldan 456 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  -> 
( F `  (
p  .x.  q )
)  e.  B )
17 opelxpi 4721 . . . . . . . . . 10  |-  ( (
<. ( F `  p
) ,  ( F `
 q ) >.  e.  ( B  X.  B
)  /\  ( F `  ( p  .x.  q
) )  e.  B
)  ->  <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>.  e.  ( ( B  X.  B )  X.  B ) )
1812, 16, 17syl2anc 642 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >.  e.  ( ( B  X.  B
)  X.  B ) )
1918snssd 3760 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  C_  ( ( B  X.  B )  X.  B
) )
2019anassrs 629 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  V )  /\  q  e.  V )  ->  { <. <.
( F `  p
) ,  ( F `
 q ) >. ,  ( F `  ( p  .x.  q ) ) >. }  C_  (
( B  X.  B
)  X.  B ) )
2120ralrimiva 2626 . . . . . 6  |-  ( (
ph  /\  p  e.  V )  ->  A. q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. }  C_  ( ( B  X.  B )  X.  B ) )
22 iunss 3943 . . . . . 6  |-  ( U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  C_  ( ( B  X.  B )  X.  B
)  <->  A. q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  C_  ( ( B  X.  B )  X.  B
) )
2321, 22sylibr 203 . . . . 5  |-  ( (
ph  /\  p  e.  V )  ->  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. }  C_  ( ( B  X.  B )  X.  B ) )
2423ralrimiva 2626 . . . 4  |-  ( ph  ->  A. p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  C_  ( ( B  X.  B )  X.  B
) )
25 iunss 3943 . . . 4  |-  ( U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. }  C_  ( ( B  X.  B )  X.  B )  <->  A. p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. }  C_  ( ( B  X.  B )  X.  B ) )
2624, 25sylibr 203 . . 3  |-  ( ph  ->  U_ p  e.  V  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  C_  ( ( B  X.  B )  X.  B
) )
273, 26eqsstrd 3212 . 2  |-  ( ph  -> 
.xb  C_  ( ( B  X.  B )  X.  B ) )
28 dff2 5672 . 2  |-  (  .xb  : ( B  X.  B
) --> B  <->  (  .xb  Fn  ( B  X.  B
)  /\  .xb  C_  (
( B  X.  B
)  X.  B ) ) )
294, 27, 28sylanbrc 645 1  |-  ( ph  -> 
.xb  : ( B  X.  B ) --> B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   {csn 3640   <.cop 3643   U_ciun 3905    X. cxp 4687    Fn wfn 5250   -->wf 5251   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858
This theorem is referenced by:  imasaddf  13435  imasmulf  13438  divsaddflem  13454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263
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