Theorem fvsnun2 5847

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5846. (Contributed by NM, 23-Sep-2007.)

Hypotheses

Ref	Expression
fvsnun.1	|- A e. _V
fvsnun.2	|- B e. _V
fvsnun.3	|- G = ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) )

Assertion

Ref	Expression
fvsnun2	|- ( D e. ( C \ { A } ) -> ( G ` D ) = ( F ` D ) )

Proof of Theorem fvsnun2

Step	Hyp	Ref	Expression
1		fvsnun.3	. . . . 5 |- G = ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) )
2	1	reseq1i 5007	. . . 4 |- ( G |` ( C \ { A } ) ) = ( ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) |` ( C \ { A } ) )
3		resundir 5025	. . . 4 |- ( ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) |` ( C \ { A } ) ) = ( ( { <. A , B >. } |` ( C \ { A } ) ) u. ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) )
4		disjdif 3565	. . . . . . 7 |- ( { A } i^i ( C \ { A } ) ) = (/)
5		fvsnun.1	. . . . . . . . 9 |- A e. _V
6		fvsnun.2	. . . . . . . . 9 |- B e. _V
7	5, 6	fnsn 5381	. . . . . . . 8 |- { <. A , B >. } Fn { A }
8		fnresdisj 5439	. . . . . . . 8 |- ( { <. A , B >. } Fn { A } -> ( ( { A } i^i ( C \ { A } ) ) = (/) <-> ( { <. A , B >. } |` ( C \ { A } ) ) = (/) ) )
9	7, 8	ax-mp 5	. . . . . . 7 |- ( ( { A } i^i ( C \ { A } ) ) = (/) <-> ( { <. A , B >. } |` ( C \ { A } ) ) = (/) )
10	4, 9	mpbi 145	. . . . . 6 |- ( { <. A , B >. } |` ( C \ { A } ) ) = (/)
11		residm 5043	. . . . . 6 |- ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) = ( F |` ( C \ { A } ) )
12	10, 11	uneq12i 3357	. . . . 5 |- ( ( { <. A , B >. } |` ( C \ { A } ) ) u. ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) ) = ( (/) u. ( F |` ( C \ { A } ) ) )
13		uncom 3349	. . . . 5 |- ( (/) u. ( F |` ( C \ { A } ) ) ) = ( ( F |` ( C \ { A } ) ) u. (/) )
14		un0 3526	. . . . 5 |- ( ( F |` ( C \ { A } ) ) u. (/) ) = ( F |` ( C \ { A } ) )
15	12, 13, 14	3eqtri 2254	. . . 4 |- ( ( { <. A , B >. } |` ( C \ { A } ) ) u. ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) ) = ( F |` ( C \ { A } ) )
16	2, 3, 15	3eqtri 2254	. . 3 |- ( G |` ( C \ { A } ) ) = ( F |` ( C \ { A } ) )
17	16	fveq1i 5636	. 2 |- ( ( G |` ( C \ { A } ) ) ` D ) = ( ( F |` ( C \ { A } ) ) ` D )
18		fvres 5659	. 2 |- ( D e. ( C \ { A } ) -> ( ( G |` ( C \ { A } ) ) ` D ) = ( G ` D ) )
19		fvres 5659	. 2 |- ( D e. ( C \ { A } ) -> ( ( F |` ( C \ { A } ) ) ` D ) = ( F ` D ) )
20	17, 18, 19	3eqtr3a 2286	1 |- ( D e. ( C \ { A } ) -> ( G ` D ) = ( F ` D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2800   \ cdif 3195   u. cun 3196   i^i cin 3197   (/)c0 3492   {csn 3667   <.cop 3670   |` cres 4725   Fn wfn 5319   ` cfv 5324

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-in1 617  ax-in2 618  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 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  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-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332

This theorem is referenced by:  facnn  10979