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Theorem fmptapd 6096
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.2 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  C  =  B )
2 fmptapd.0a . . . 4  |-  ( ph  ->  A  e.  _V )
3 fmptapd.0b . . . 4  |-  ( ph  ->  B  e.  _V )
41, 2, 3fmptsnd 6094 . . 3  |-  ( ph  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  C ) )
54uneq2d 3654 . 2  |-  ( ph  ->  ( ( x  e.  R  |->  C )  u. 
{ <. A ,  B >. } )  =  ( ( x  e.  R  |->  C )  u.  (
x  e.  { A }  |->  C ) ) )
6 mptun 5718 . . 3  |-  ( x  e.  ( R  u.  { A } )  |->  C )  =  ( ( x  e.  R  |->  C )  u.  ( x  e.  { A }  |->  C ) )
76a1i 11 . 2  |-  ( ph  ->  ( x  e.  ( R  u.  { A } )  |->  C )  =  ( ( x  e.  R  |->  C )  u.  ( x  e. 
{ A }  |->  C ) ) )
8 fmptapd.1 . . 3  |-  ( ph  ->  ( R  u.  { A } )  =  S )
98mpteq1d 4538 . 2  |-  ( ph  ->  ( x  e.  ( R  u.  { A } )  |->  C )  =  ( x  e.  S  |->  C ) )
105, 7, 93eqtr2d 2504 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 1395    e. wcel 1819   _Vcvv 3109    u. cun 3469   {csn 4032   <.cop 4038    |-> cmpt 4515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516  df-mpt 4517
This theorem is referenced by:  fmptpr  6097
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