Theorem fvsnun2 5805

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 5804. (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 4974	. . . 4 |- ( G |` ( C \ { A } ) ) = ( ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) |` ( C \ { A } ) )
3		resundir 4992	. . . 4 |- ( ( { <. A , B >. } u. ( F |` ( C \ { A } ) ) ) |` ( C \ { A } ) ) = ( ( { <. A , B >. } |` ( C \ { A } ) ) u. ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) )
4		disjdif 3541	. . . . . . 7 |- ( { A } i^i ( C \ { A } ) ) = (/)
5		fvsnun.1	. . . . . . . . 9 |- A e. _V
6		fvsnun.2	. . . . . . . . 9 |- B e. _V
7	5, 6	fnsn 5347	. . . . . . . 8 |- { <. A , B >. } Fn { A }
8		fnresdisj 5405	. . . . . . . 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 5010	. . . . . 6 |- ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) = ( F |` ( C \ { A } ) )
12	10, 11	uneq12i 3333	. . . . 5 |- ( ( { <. A , B >. } |` ( C \ { A } ) ) u. ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) ) = ( (/) u. ( F |` ( C \ { A } ) ) )
13		uncom 3325	. . . . 5 |- ( (/) u. ( F |` ( C \ { A } ) ) ) = ( ( F |` ( C \ { A } ) ) u. (/) )
14		un0 3502	. . . . 5 |- ( ( F |` ( C \ { A } ) ) u. (/) ) = ( F |` ( C \ { A } ) )
15	12, 13, 14	3eqtri 2232	. . . 4 |- ( ( { <. A , B >. } |` ( C \ { A } ) ) u. ( ( F |` ( C \ { A } ) ) |` ( C \ { A } ) ) ) = ( F |` ( C \ { A } ) )
16	2, 3, 15	3eqtri 2232	. . 3 |- ( G |` ( C \ { A } ) ) = ( F |` ( C \ { A } ) )
17	16	fveq1i 5600	. 2 |- ( ( G |` ( C \ { A } ) ) ` D ) = ( ( F |` ( C \ { A } ) ) ` D )
18		fvres 5623	. 2 |- ( D e. ( C \ { A } ) -> ( ( G |` ( C \ { A } ) ) ` D ) = ( G ` D ) )
19		fvres 5623	. 2 |- ( D e. ( C \ { A } ) -> ( ( F |` ( C \ { A } ) ) ` D ) = ( F ` D ) )
20	17, 18, 19	3eqtr3a 2264	1 |- ( D e. ( C \ { A } ) -> ( G ` D ) = ( F ` D ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1373   e. wcel 2178   _Vcvv 2776   \ cdif 3171   u. cun 3172   i^i cin 3173   (/)c0 3468   {csn 3643   <.cop 3646   |` cres 4695   Fn wfn 5285   ` cfv 5290

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298

This theorem is referenced by:  facnn  10909