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Theorem fnprOLD 6115
Description: Obsolete version of fnprb 6114 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.) (New usage is discouraged.) (Proof modification is discouraged.)
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 5670 . . . . 5  |-  ( F  Fn  { I ,  J }  ->  dom  F  =  { I ,  J } )
2 fvex 5866 . . . . . 6  |-  ( F `
 I )  e. 
_V
3 fvex 5866 . . . . . 6  |-  ( F `
 J )  e. 
_V
42, 3dmprop 5473 . . . . 5  |-  dom  { <. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. }  =  { I ,  J }
51, 4syl6eqr 2502 . . . 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 2513 . . . . 5  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  e.  dom  F  <-> 
x  e.  { I ,  J } ) )
9 vex 3098 . . . . . . 7  |-  x  e. 
_V
109elpr 4032 . . . . . 6  |-  ( x  e.  { I ,  J }  <->  ( x  =  I  \/  x  =  J ) )
11 fnprOLD.1 . . . . . . . . . . 11  |-  I  e. 
_V
1211, 2fvpr1 6099 . . . . . . . . . 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 2451 . . . . . . . 8  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( F `  I
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  I ) )
15 fveq2 5856 . . . . . . . . 9  |-  ( x  =  I  ->  ( F `  x )  =  ( F `  I ) )
16 fveq2 5856 . . . . . . . . 9  |-  ( x  =  I  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  I ) )
1715, 16eqeq12d 2465 . . . . . . . 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 6100 . . . . . . . . . 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 2451 . . . . . . . 8  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( F `  J
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  J ) )
23 fveq2 5856 . . . . . . . . 9  |-  ( x  =  J  ->  ( F `  x )  =  ( F `  J ) )
24 fveq2 5856 . . . . . . . . 9  |-  ( x  =  J  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  J ) )
2523, 24eqeq12d 2465 . . . . . . . 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 2855 . . 3  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  ->  A. x  e.  dom  F ( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x ) )
31 fnfun 5668 . . . 4  |-  ( F  Fn  { I ,  J }  ->  Fun  F )
3211, 19, 2, 3funpr 5629 . . . 4  |-  ( I  =/=  J  ->  Fun  {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } )
33 eqfunfv 5971 . . . 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 922 . 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 5581 . . . 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 5659 . . . 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 832 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 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   _Vcvv 3095   {cpr 4016   <.cop 4020   dom cdm 4989   Fun wfun 5572    Fn wfn 5573   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586
This theorem is referenced by: (None)
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