Theorem fmptapd 5830

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

Step	Hyp	Ref	Expression
1		fmptapd.0a	. . . . 5 |- ( ph -> A e. _V )
2		fmptapd.0b	. . . . 5 |- ( ph -> B e. _V )
3		fmptsn 5828	. . . . 5 |- ( ( A e. _V /\ B e. _V ) -> { <. A , B >. } = ( x e. { A } |-> B ) )
4	1, 2, 3	syl2anc 411	. . . 4 |- ( ph -> { <. A , B >. } = ( x e. { A } |-> B ) )
5		elsni 3684	. . . . . 6 |- ( x e. { A } -> x = A )
6		fmptapd.2	. . . . . 6 |- ( ( ph /\ x = A ) -> C = B )
7	5, 6	sylan2 286	. . . . 5 |- ( ( ph /\ x e. { A } ) -> C = B )
8	7	mpteq2dva 4174	. . . 4 |- ( ph -> ( x e. { A } |-> C ) = ( x e. { A } |-> B ) )
9	4, 8	eqtr4d 2265	. . 3 |- ( ph -> { <. A , B >. } = ( x e. { A } |-> C ) )
10	9	uneq2d 3358	. 2 |- ( ph -> ( ( x e. R |-> C ) u. { <. A , B >. } ) = ( ( x e. R |-> C ) u. ( x e. { A } |-> C ) ) )
11		mptun 5455	. . 3 |- ( x e. ( R u. { A } ) |-> C ) = ( ( x e. R |-> C ) u. ( x e. { A } |-> C ) )
12	11	a1i 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 )
14	13	mpteq1d 4169	. 2 |- ( ph -> ( x e. ( R u. { A } ) |-> C ) = ( x e. S |-> C ) )
15	10, 12, 14	3eqtr2d 2268	1 |- ( ph -> ( ( x e. R |-> C ) u. { <. A , B >. } ) = ( x e. S |-> C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   _Vcvv 2799   u. cun 3195   {csn 3666   <.cop 3669   |-> cmpt 4145

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325

This theorem is referenced by:  fmptpr  5831