Theorem fvsnun2 5361

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5360. (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 4608	. . . 4 |- ( G |` ( C \ { A } ) ) = ( ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) |` ( C \ { A } ) )
3		resundir 4626	. . . 4 |- ( ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) |` ( C \ { A } ) ) = ( ( { <. A , B >. } |` ( C \ { A } ) ) u. ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) )
4		disjdif 3296	. . . . . . 7 |- ( { A } i^i ( C \ { A } ) ) = (/)
5		fvsnun.1	. . . . . . . . 9 |- A e. _V
6		fvsnun.2	. . . . . . . . 9 |- B e. _V
7	5, 6	fnsn 4953	. . . . . . . 8 |- { <. A , B >. } Fn { A }
8		fnresdisj 5009	. . . . . . . 8 |- ( { <. A , B >. } Fn { A } -> ( ( { A } i^i ( C \ { A } ) ) = (/) <-> ( { <. A , B >. } |` ( C \ { A } ) ) = (/) ) )
9	7, 8	ax-mp 7	. . . . . . 7 |- ( ( { A } i^i ( C \ { A } ) ) = (/) <-> ( { <. A , B >. } |` ( C \ { A } ) ) = (/) )
10	4, 9	mpbi 133	. . . . . 6 |- ( { <. A , B >. } |` ( C \ { A } ) ) = (/)
11		residm 4642	. . . . . 6 |- ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) = ( F |` ( C \ { A } ) )
12	10, 11	uneq12i 3095	. . . . 5 |- ( ( { <. A , B >. } |` ( C \ { A } ) ) u. ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) ) = ( (/) u. ( F |` ( C \ { A } ) ) )
13		uncom 3087	. . . . 5 |- ( (/) u. ( F |` ( C \ { A } ) ) ) = ( ( F |` ( C \ { A } ) ) u. (/) )
14		un0 3251	. . . . 5 |- ( ( F |` ( C \ { A } ) ) u. (/) ) = ( F |` ( C \ { A } ) )
15	12, 13, 14	3eqtri 2064	. . . 4 |- ( ( { <. A , B >. } |` ( C \ { A } ) ) u. ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) ) = ( F |` ( C \ { A } ) )
16	2, 3, 15	3eqtri 2064	. . 3 |- ( G |` ( C \ { A } ) ) = ( F |` ( C \ { A } ) )
17	16	fveq1i 5179	. 2 |- ( ( G |` ( C \ { A } ) ) ` D ) = ( ( F |` ( C \ { A } ) ) ` D )
18		fvres 5198	. 2 |- ( D e. ( C \ { A } ) -> ( ( G |` ( C \ { A } ) ) ` D ) = ( G ` D ) )
19		fvres 5198	. 2 |- ( D e. ( C \ { A } ) -> ( ( F |` ( C \ { A } ) ) ` D ) = ( F ` D ) )
20	17, 18, 19	3eqtr3a 2096	1 |- ( D e. ( C \ { A } ) -> ( G ` D ) = ( F ` D ) )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   \ cdif 2914   u. cun 2915   i^i cin 2916   (/)c0 3224   {csn 3375   <.cop 3378   |` cres 4347   Fn wfn 4897   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-res 4357  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910

This theorem is referenced by: (None)