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Theorem fmptapd 5902
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 5899 . . . . 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 3902 . . . . . 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 4378 . . . 4  |-  ( ph  ->  ( x  e.  { A }  |->  C )  =  ( x  e. 
{ A }  |->  B ) )
94, 8eqtr4d 2478 . . 3  |-  ( ph  ->  { <. A ,  B >. }  =  ( x  e.  { A }  |->  C ) )
109uneq2d 3510 . 2  |-  ( ph  ->  ( ( x  e.  R  |->  C )  u. 
{ <. A ,  B >. } )  =  ( ( x  e.  R  |->  C )  u.  (
x  e.  { A }  |->  C ) ) )
11 mptun 5541 . . 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 4373 . 2  |-  ( ph  ->  ( x  e.  ( R  u.  { A } )  |->  C )  =  ( x  e.  S  |->  C ) )
1510, 12, 143eqtr2d 2481 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 1369    e. wcel 1756   _Vcvv 2972    u. cun 3326   {csn 3877   <.cop 3883    e. cmpt 4350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425
This theorem is referenced by:  fmptpr  5903
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