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Theorem imasaddflem 14776
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 14774 . 2  |-  ( ph  -> 
.xb  Fn  ( B  X.  B ) )
5 fof 5788 . . . . . . . . . . . 12  |-  ( F : V -onto-> B  ->  F : V --> B )
61, 5syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F : V --> B )
7 ffvelrn 6012 . . . . . . . . . . . . 13  |-  ( ( F : V --> B  /\  p  e.  V )  ->  ( F `  p
)  e.  B )
8 ffvelrn 6012 . . . . . . . . . . . . 13  |-  ( ( F : V --> B  /\  q  e.  V )  ->  ( F `  q
)  e.  B )
97, 8anim12dan 834 . . . . . . . . . . . 12  |-  ( ( F : V --> B  /\  ( p  e.  V  /\  q  e.  V
) )  ->  (
( F `  p
)  e.  B  /\  ( F `  q )  e.  B ) )
10 opelxpi 5025 . . . . . . . . . . . 12  |-  ( ( ( F `  p
)  e.  B  /\  ( F `  q )  e.  B )  ->  <. ( F `  p
) ,  ( F `
 q ) >.  e.  ( B  X.  B
) )
119, 10syl 16 . . . . . . . . . . 11  |-  ( ( F : V --> B  /\  ( p  e.  V  /\  q  e.  V
) )  ->  <. ( F `  p ) ,  ( F `  q ) >.  e.  ( B  X.  B ) )
126, 11sylan 471 . . . . . . . . . 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 6012 . . . . . . . . . . . 12  |-  ( ( F : V --> B  /\  ( p  .x.  q )  e.  V )  -> 
( F `  (
p  .x.  q )
)  e.  B )
156, 14sylan 471 . . . . . . . . . . 11  |-  ( (
ph  /\  ( p  .x.  q )  e.  V
)  ->  ( F `  ( p  .x.  q
) )  e.  B
)
1613, 15syldan 470 . . . . . . . . . 10  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  -> 
( F `  (
p  .x.  q )
)  e.  B )
17 opelxpi 5025 . . . . . . . . . 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 661 . . . . . . . . 9  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >.  e.  ( ( B  X.  B
)  X.  B ) )
1918snssd 4167 . . . . . . . 8  |-  ( (
ph  /\  ( p  e.  V  /\  q  e.  V ) )  ->  { <. <. ( F `  p ) ,  ( F `  q )
>. ,  ( F `  ( p  .x.  q
) ) >. }  C_  ( ( B  X.  B )  X.  B
) )
2019anassrs 648 . . . . . . 7  |-  ( ( ( ph  /\  p  e.  V )  /\  q  e.  V )  ->  { <. <.
( F `  p
) ,  ( F `
 q ) >. ,  ( F `  ( p  .x.  q ) ) >. }  C_  (
( B  X.  B
)  X.  B ) )
2120ralrimiva 2873 . . . . . 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 4361 . . . . . 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 212 . . . . 5  |-  ( (
ph  /\  p  e.  V )  ->  U_ q  e.  V  { <. <. ( F `  p ) ,  ( F `  q ) >. ,  ( F `  ( p 
.x.  q ) )
>. }  C_  ( ( B  X.  B )  X.  B ) )
2423ralrimiva 2873 . . . 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 4361 . . . 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 212 . . 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 3533 . 2  |-  ( ph  -> 
.xb  C_  ( ( B  X.  B )  X.  B ) )
28 dff2 6026 . 2  |-  (  .xb  : ( B  X.  B
) --> B  <->  (  .xb  Fn  ( B  X.  B
)  /\  .xb  C_  (
( B  X.  B
)  X.  B ) ) )
294, 27, 28sylanbrc 664 1  |-  ( ph  -> 
.xb  : ( B  X.  B ) --> B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   A.wral 2809    C_ wss 3471   {csn 4022   <.cop 4028   U_ciun 4320    X. cxp 4992    Fn wfn 5576   -->wf 5577   -onto->wfo 5579   ` cfv 5581  (class class class)co 6277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-fo 5587  df-fv 5589
This theorem is referenced by:  imasaddf  14779  imasmulf  14782  divsaddflem  14798
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