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Theorem repfuntw 25160
Description: Representation as a set of pairs of a function whose domain has two distinct elements. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.)
Hypotheses
Ref Expression
repfuntw.1  |-  I  e.  A
repfuntw.2  |-  J  e.  B
Assertion
Ref Expression
repfuntw  |-  ( I  =/=  J  ->  ( F  Fn  { I ,  J }  <->  F  =  { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } ) )

Proof of Theorem repfuntw
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fndm 5343 . . . . 5  |-  ( F  Fn  { I ,  J }  ->  dom  F  =  { I ,  J } )
2 fvex 5539 . . . . . 6  |-  ( F `
 I )  e. 
_V
3 fvex 5539 . . . . . 6  |-  ( F `
 J )  e. 
_V
42, 3dmprop 5148 . . . . 5  |-  dom  { <. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. }  =  { I ,  J }
51, 4syl6eqr 2333 . . . 4  |-  ( F  Fn  { I ,  J }  ->  dom  F  =  dom  { <. I ,  ( F `  I ) >. ,  <. J ,  ( F `  J ) >. } )
65adantl 452 . . 3  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  ->  dom  F  =  dom  { <. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } )
71adantl 452 . . . . . 6  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  ->  dom  F  =  { I ,  J } )
87eleq2d 2350 . . . . 5  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  e.  dom  F  <-> 
x  e.  { I ,  J } ) )
9 vex 2791 . . . . . . 7  |-  x  e. 
_V
109elpr 3658 . . . . . 6  |-  ( x  e.  { I ,  J }  <->  ( x  =  I  \/  x  =  J ) )
11 repfuntw.1 . . . . . . . . . . . 12  |-  I  e.  A
1211elexi 2797 . . . . . . . . . . 11  |-  I  e. 
_V
1312, 2fvpr1 5722 . . . . . . . . . 10  |-  ( I  =/=  J  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  I
)  =  ( F `
 I ) )
1413adantr 451 . . . . . . . . 9  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. } `  I
)  =  ( F `
 I ) )
1514eqcomd 2288 . . . . . . . 8  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( F `  I
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  I ) )
16 fveq2 5525 . . . . . . . . 9  |-  ( x  =  I  ->  ( F `  x )  =  ( F `  I ) )
17 fveq2 5525 . . . . . . . . 9  |-  ( x  =  I  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  I ) )
1816, 17eqeq12d 2297 . . . . . . . 8  |-  ( x  =  I  ->  (
( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x )  <-> 
( F `  I
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  I ) ) )
1915, 18syl5ibrcom 213 . . . . . . 7  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  =  I  ->  ( F `  x )  =  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
) ) )
20 repfuntw.2 . . . . . . . . . . . 12  |-  J  e.  B
2120elexi 2797 . . . . . . . . . . 11  |-  J  e. 
_V
2221, 3fvpr2 5723 . . . . . . . . . 10  |-  ( I  =/=  J  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  J
)  =  ( F `
 J ) )
2322adantr 451 . . . . . . . . 9  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. } `  J
)  =  ( F `
 J ) )
2423eqcomd 2288 . . . . . . . 8  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( F `  J
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  J ) )
25 fveq2 5525 . . . . . . . . 9  |-  ( x  =  J  ->  ( F `  x )  =  ( F `  J ) )
26 fveq2 5525 . . . . . . . . 9  |-  ( x  =  J  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  J ) )
2725, 26eqeq12d 2297 . . . . . . . 8  |-  ( x  =  J  ->  (
( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x )  <-> 
( F `  J
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  J ) ) )
2824, 27syl5ibrcom 213 . . . . . . 7  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  =  J  ->  ( F `  x )  =  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
) ) )
2919, 28jaod 369 . . . . . 6  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( ( x  =  I  \/  x  =  J )  ->  ( F `  x )  =  ( { <. I ,  ( F `  I ) >. ,  <. J ,  ( F `  J ) >. } `  x ) ) )
3010, 29syl5bi 208 . . . . 5  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  e.  {
I ,  J }  ->  ( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x ) ) )
318, 30sylbid 206 . . . 4  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  e.  dom  F  ->  ( F `  x )  =  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
) ) )
3231ralrimiv 2625 . . 3  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  ->  A. x  e.  dom  F ( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x ) )
33 fnfun 5341 . . . 4  |-  ( F  Fn  { I ,  J }  ->  Fun  F )
3412, 21, 2, 3funpr 5302 . . . 4  |-  ( I  =/=  J  ->  Fun  {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } )
35 eqfunfv 5627 . . . 4  |-  ( ( Fun  F  /\  Fun  {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } )  ->  ( F  =  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. }  <->  ( dom  F  =  dom  { <. I ,  ( F `  I ) >. ,  <. J ,  ( F `  J ) >. }  /\  A. x  e.  dom  F
( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x ) ) ) )
3633, 34, 35syl2anr 464 . . 3  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( F  =  { <. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. }  <->  ( dom  F  =  dom  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. }  /\  A. x  e.  dom  F ( F `  x )  =  ( { <. I ,  ( F `  I ) >. ,  <. J ,  ( F `  J ) >. } `  x ) ) ) )
376, 32, 36mpbir2and 888 . 2  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  ->  F  =  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. } )
384a1i 10 . . . 4  |-  ( I  =/=  J  ->  dom  {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. }  =  { I ,  J } )
39 df-fn 5258 . . . 4  |-  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. }  Fn  { I ,  J }  <->  ( Fun  {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. }  /\  dom  { <. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. }  =  { I ,  J } ) )
4034, 38, 39sylanbrc 645 . . 3  |-  ( I  =/=  J  ->  { <. I ,  ( F `  I ) >. ,  <. J ,  ( F `  J ) >. }  Fn  { I ,  J }
)
41 fneq1 5333 . . . 4  |-  ( F  =  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. }  ->  ( F  Fn  { I ,  J }  <->  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. }  Fn  {
I ,  J }
) )
4241biimprd 214 . . 3  |-  ( F  =  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. }  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. }  Fn  {
I ,  J }  ->  F  Fn  { I ,  J } ) )
4340, 42mpan9 455 . 2  |-  ( ( I  =/=  J  /\  F  =  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. } )  ->  F  Fn  { I ,  J } )
4437, 43impbida 805 1  |-  ( I  =/=  J  ->  ( F  Fn  { I ,  J }  <->  F  =  { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   {cpr 3641   <.cop 3643   dom cdm 4689   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  repcpwti  25161
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-13 1686  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-pow 4188  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-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-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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