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Theorem fmptapd 6083
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 6079 . . . . 5  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  B ) )
41, 2, 3syl2anc 661 . . . 4  |-  ( ph  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  B ) )
5 elsni 4052 . . . . . 6  |-  ( x  e.  { A }  ->  x  =  A )
6 fmptapd.2 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  C  =  B )
75, 6sylan2 474 . . . . 5  |-  ( (
ph  /\  x  e.  { A } )  ->  C  =  B )
87mpteq2dva 4533 . . . 4  |-  ( ph  ->  ( x  e.  { A }  |->  C )  =  ( x  e. 
{ A }  |->  B ) )
94, 8eqtr4d 2511 . . 3  |-  ( ph  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  C ) )
109uneq2d 3658 . 2  |-  ( ph  ->  ( ( x  e.  R  |->  C )  u. 
{ <. A ,  B >. } )  =  ( ( x  e.  R  |->  C )  u.  (
x  e.  { A }  |->  C ) ) )
11 mptun 5710 . . 3  |-  ( x  e.  ( R  u.  { A } )  |->  C )  =  ( ( x  e.  R  |->  C )  u.  ( x  e.  { A }  |->  C ) )
1211a1i 11 . 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 4528 . 2  |-  ( ph  ->  ( x  e.  ( R  u.  { A } )  |->  C )  =  ( x  e.  S  |->  C ) )
1510, 12, 143eqtr2d 2514 1  |-  ( ph  ->  ( ( x  e.  R  |->  C )  u. 
{ <. A ,  B >. } )  =  ( x  e.  S  |->  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    u. cun 3474   {csn 4027   <.cop 4033    |-> cmpt 4505
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  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-reu 2821  df-rab 2823  df-v 3115  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-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-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593
This theorem is referenced by:  fmptpr  6084
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