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Theorem fnprOLD 6120
Description: Obsolete version of fnprb 6119 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 5680 . . . . 5  |-  ( F  Fn  { I ,  J }  ->  dom  F  =  { I ,  J } )
2 fvex 5876 . . . . . 6  |-  ( F `
 I )  e. 
_V
3 fvex 5876 . . . . . 6  |-  ( F `
 J )  e. 
_V
42, 3dmprop 5483 . . . . 5  |-  dom  { <. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. }  =  { I ,  J }
51, 4syl6eqr 2526 . . . 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 2537 . . . . 5  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( x  e.  dom  F  <-> 
x  e.  { I ,  J } ) )
9 vex 3116 . . . . . . 7  |-  x  e. 
_V
109elpr 4045 . . . . . 6  |-  ( x  e.  { I ,  J }  <->  ( x  =  I  \/  x  =  J ) )
11 fnprOLD.1 . . . . . . . . . . 11  |-  I  e. 
_V
1211, 2fvpr1 6104 . . . . . . . . . 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 2475 . . . . . . . 8  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( F `  I
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  I ) )
15 fveq2 5866 . . . . . . . . 9  |-  ( x  =  I  ->  ( F `  x )  =  ( F `  I ) )
16 fveq2 5866 . . . . . . . . 9  |-  ( x  =  I  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  I ) )
1715, 16eqeq12d 2489 . . . . . . . 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 6105 . . . . . . . . . 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 2475 . . . . . . . 8  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  -> 
( F `  J
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  J ) )
23 fveq2 5866 . . . . . . . . 9  |-  ( x  =  J  ->  ( F `  x )  =  ( F `  J ) )
24 fveq2 5866 . . . . . . . . 9  |-  ( x  =  J  ->  ( { <. I ,  ( F `  I )
>. ,  <. J , 
( F `  J
) >. } `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  J ) )
2523, 24eqeq12d 2489 . . . . . . . 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 2876 . . 3  |-  ( ( I  =/=  J  /\  F  Fn  { I ,  J } )  ->  A. x  e.  dom  F ( F `  x
)  =  ( {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } `  x ) )
31 fnfun 5678 . . . 4  |-  ( F  Fn  { I ,  J }  ->  Fun  F )
3211, 19, 2, 3funpr 5639 . . . 4  |-  ( I  =/=  J  ->  Fun  {
<. I ,  ( F `
 I ) >. ,  <. J ,  ( F `  J )
>. } )
33 eqfunfv 5980 . . . 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 920 . 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 5591 . . . 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 5669 . . . 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 830 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 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   _Vcvv 3113   {cpr 4029   <.cop 4033   dom cdm 4999   Fun wfun 5582    Fn wfn 5583   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-fv 5596
This theorem is referenced by: (None)
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