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Theorem fnprOLD 5942
Description: Obsolete version of fnprb 5941 as of 29-Dec-2018. 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.) (Revised by NM, 10-Dec-2017.)
Hypotheses
Ref Expression
fnprOLD.1  |-  I  e. 
_V
fnprOLD.2  |-  J  e. 
_V
Assertion
Ref Expression
fnprOLD  |-  ( I  =/=  J  ->  ( F  Fn  { I ,  J }  <->  F  =  { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } ) )

Proof of Theorem fnprOLD
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fndm 5515 . . . . 5  |-  ( F  Fn  { I ,  J }  ->  dom  F  =  { I ,  J } )
2 fvex 5706 . . . . . 6  |-  ( F `
 I )  e. 
_V
3 fvex 5706 . . . . . 6  |-  ( F `
 J )  e. 
_V
42, 3dmprop 5319 . . . . 5  |-  dom  { <. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. }  =  { I ,  J }
51, 4syl6eqr 2493 . . . 4  |-  ( F  Fn  { I ,  J }  ->  dom  F  =  dom  { <. I ,  ( F `  I ) >. ,  <. J ,  ( F `  J ) >. } )
65adantl 466 . . 3  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  ->  dom  F  =  dom  { <. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } )
71adantl 466 . . . . . 6  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  ->  dom  F  =  { I ,  J } )
87eleq2d 2510 . . . . 5  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  e.  dom  F  <-> 
x  e.  { I ,  J } ) )
9 vex 2980 . . . . . . 7  |-  x  e. 
_V
109elpr 3900 . . . . . 6  |-  ( x  e.  { I ,  J }  <->  ( x  =  I  \/  x  =  J ) )
11 fnprOLD.1 . . . . . . . . . . 11  |-  I  e. 
_V
1211, 2fvpr1 5926 . . . . . . . . . 10  |-  ( I  =/=  J  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  I
)  =  ( F `
 I ) )
1312adantr 465 . . . . . . . . 9  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. } `  I
)  =  ( F `
 I ) )
1413eqcomd 2448 . . . . . . . 8  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( F `  I
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  I ) )
15 fveq2 5696 . . . . . . . . 9  |-  ( x  =  I  ->  ( F `  x )  =  ( F `  I ) )
16 fveq2 5696 . . . . . . . . 9  |-  ( x  =  I  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  I ) )
1715, 16eqeq12d 2457 . . . . . . . 8  |-  ( x  =  I  ->  (
( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x )  <-> 
( F `  I
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  I ) ) )
1814, 17syl5ibrcom 222 . . . . . . 7  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  =  I  ->  ( F `  x )  =  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
) ) )
19 fnprOLD.2 . . . . . . . . . . 11  |-  J  e. 
_V
2019, 3fvpr2 5927 . . . . . . . . . 10  |-  ( I  =/=  J  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  J
)  =  ( F `
 J ) )
2120adantr 465 . . . . . . . . 9  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. } `  J
)  =  ( F `
 J ) )
2221eqcomd 2448 . . . . . . . 8  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( F `  J
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  J ) )
23 fveq2 5696 . . . . . . . . 9  |-  ( x  =  J  ->  ( F `  x )  =  ( F `  J ) )
24 fveq2 5696 . . . . . . . . 9  |-  ( x  =  J  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  J ) )
2523, 24eqeq12d 2457 . . . . . . . 8  |-  ( x  =  J  ->  (
( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x )  <-> 
( F `  J
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  J ) ) )
2622, 25syl5ibrcom 222 . . . . . . 7  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  =  J  ->  ( F `  x )  =  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
) ) )
2718, 26jaod 380 . . . . . 6  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( ( x  =  I  \/  x  =  J )  ->  ( F `  x )  =  ( { <. I ,  ( F `  I ) >. ,  <. J ,  ( F `  J ) >. } `  x ) ) )
2810, 27syl5bi 217 . . . . 5  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  e.  {
I ,  J }  ->  ( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x ) ) )
298, 28sylbid 215 . . . 4  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  e.  dom  F  ->  ( F `  x )  =  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
) ) )
3029ralrimiv 2803 . . 3  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  ->  A. x  e.  dom  F ( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x ) )
31 fnfun 5513 . . . 4  |-  ( F  Fn  { I ,  J }  ->  Fun  F )
3211, 19, 2, 3funpr 5474 . . . 4  |-  ( I  =/=  J  ->  Fun  {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } )
33 eqfunfv 5807 . . . 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 ) ) ) )
3431, 32, 33syl2anr 478 . . 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 ) ) ) )
356, 30, 34mpbir2and 913 . 2  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  ->  F  =  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. } )
364a1i 11 . . . 4  |-  ( I  =/=  J  ->  dom  {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. }  =  { I ,  J } )
37 df-fn 5426 . . . 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 } ) )
3832, 36, 37sylanbrc 664 . . 3  |-  ( I  =/=  J  ->  { <. I ,  ( F `  I ) >. ,  <. J ,  ( F `  J ) >. }  Fn  { I ,  J }
)
39 fneq1 5504 . . . 4  |-  ( F  =  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. }  ->  ( F  Fn  { I ,  J }  <->  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. }  Fn  {
I ,  J }
) )
4039biimprd 223 . . 3  |-  ( F  =  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. }  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. }  Fn  {
I ,  J }  ->  F  Fn  { I ,  J } ) )
4138, 40mpan9 469 . 2  |-  ( ( I  =/=  J  /\  F  =  { <. I ,  ( F `  I
) >. ,  <. J , 
( F `  J
) >. } )  ->  F  Fn  { I ,  J } )
4235, 41impbida 828 1  |-  ( I  =/=  J  ->  ( F  Fn  { I ,  J }  <->  F  =  { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   _Vcvv 2977   {cpr 3884   <.cop 3888   dom cdm 4845   Fun wfun 5417    Fn wfn 5418   ` cfv 5423
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 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431
This theorem is referenced by: (None)
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