ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fmptapd Unicode version

Theorem fmptapd 5472
Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)
Hypotheses
Ref Expression
fmptapd.0a  |-  ( ph  ->  A  e.  _V )
fmptapd.0b  |-  ( ph  ->  B  e.  _V )
fmptapd.1  |-  ( ph  ->  ( R  u.  { A } )  =  S )
fmptapd.2  |-  ( (
ph  /\  x  =  A )  ->  C  =  B )
Assertion
Ref Expression
fmptapd  |-  ( ph  ->  ( ( x  e.  R  |->  C )  u. 
{ <. A ,  B >. } )  =  ( x  e.  S  |->  C ) )
Distinct variable groups:    x, A    x, B    x, R    x, S    ph, x
Allowed substitution hint:    C( x)

Proof of Theorem fmptapd
StepHypRef Expression
1 fmptapd.0a . . . . 5  |-  ( ph  ->  A  e.  _V )
2 fmptapd.0b . . . . 5  |-  ( ph  ->  B  e.  _V )
3 fmptsn 5470 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  B ) )
41, 2, 3syl2anc 403 . . . 4  |-  ( ph  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  B ) )
5 elsni 3459 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
6 fmptapd.2 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  C  =  B )
75, 6sylan2 280 . . . . 5  |-  ( (
ph  /\  x  e.  { A } )  ->  C  =  B )
87mpteq2dva 3920 . . . 4  |-  ( ph  ->  ( x  e.  { A }  |->  C )  =  ( x  e. 
{ A }  |->  B ) )
94, 8eqtr4d 2123 . . 3  |-  ( ph  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  C ) )
109uneq2d 3152 . 2  |-  ( ph  ->  ( ( x  e.  R  |->  C )  u. 
{ <. A ,  B >. } )  =  ( ( x  e.  R  |->  C )  u.  (
x  e.  { A }  |->  C ) ) )
11 mptun 5130 . . 3  |-  ( x  e.  ( R  u.  { A } )  |->  C )  =  ( ( x  e.  R  |->  C )  u.  ( x  e.  { A }  |->  C ) )
1211a1i 9 . 2  |-  ( ph  ->  ( x  e.  ( R  u.  { A } )  |->  C )  =  ( ( x  e.  R  |->  C )  u.  ( x  e. 
{ A }  |->  C ) ) )
13 fmptapd.1 . . 3  |-  ( ph  ->  ( R  u.  { A } )  =  S )
1413mpteq1d 3915 . 2  |-  ( ph  ->  ( x  e.  ( R  u.  { A } )  |->  C )  =  ( x  e.  S  |->  C ) )
1510, 12, 143eqtr2d 2126 1  |-  ( ph  ->  ( ( x  e.  R  |->  C )  u. 
{ <. A ,  B >. } )  =  ( x  e.  S  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619    u. cun 2995   {csn 3441   <.cop 3444    |-> cmpt 3891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-mpt 3893  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009
This theorem is referenced by:  fmptpr  5473
  Copyright terms: Public domain W3C validator