Theorem fvsnun2 4765

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 4764.

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		fvres 4691	. 2 |- (D e. (C \ {A}) -> ((G |` (C \ {A}))` D) = (G` D))
2		fvres 4691	. . 3 |- (D e. (C \ {A}) -> ((F |` (C \ {A}))` D) = (F` D))
3		fvsnun.3	. . . . . 6 |- G = ({<.A, B>.} u. (F |` (C \ {A})))
4		reseq1 4218	. . . . . 6 |- (G = ({<.A, B>.} u. (F |` (C \ {A}))) -> (G |` (C \ {A})) = (({<.A, B>.} u. (F |` (C \ {A}))) |` (C \ {A})))
5	3, 4	ax-mp 7	. . . . 5 |- (G |` (C \ {A})) = (({<.A, B>.} u. (F |` (C \ {A}))) |` (C \ {A}))
6		resundir 4230	. . . . 5 |- (({<.A, B>.} u. (F |` (C \ {A}))) |` (C \ {A})) = (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A})))
7		difdisj 2945	. . . . . . . 8 |- ({A} i^i (C \ {A})) = (/)
8		fvsnun.1	. . . . . . . . . . 11 |- A e. _V
9		fvsnun.2	. . . . . . . . . . 11 |- B e. _V
10	8, 9	f1osn 4674	. . . . . . . . . 10 |- {<.A, B>.}:{A}-1-1-onto->{B}
11		f1ofn 4636	. . . . . . . . . 10 |- ({<.A, B>.}:{A}-1-1-onto->{B} -> {<.A, B>.} Fn {A})
12	10, 11	ax-mp 7	. . . . . . . . 9 |- {<.A, B>.} Fn {A}
13		fnresdisj 4523	. . . . . . . . 9 |- ({<.A, B>.} Fn {A} -> (({A} i^i (C \ {A})) = (/) <-> ({<.A, B>.} |` (C \ {A})) = (/)))
14	12, 13	ax-mp 7	. . . . . . . 8 |- (({A} i^i (C \ {A})) = (/) <-> ({<.A, B>.} |` (C \ {A})) = (/))
15	7, 14	mpbi 206	. . . . . . 7 |- ({<.A, B>.} |` (C \ {A})) = (/)
16		residm 4246	. . . . . . 7 |- ((F |` (C \ {A})) |` (C \ {A})) = (F |` (C \ {A}))
17	15, 16	uneq12i 2753	. . . . . 6 |- (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A}))) = ((/) u. (F |` (C \ {A})))
18		uncom 2744	. . . . . 6 |- ((/) u. (F |` (C \ {A}))) = ((F |` (C \ {A})) u. (/))
19		un0 2896	. . . . . 6 |- ((F |` (C \ {A})) u. (/)) = (F |` (C \ {A}))
20	17, 18, 19	3eqtri 1912	. . . . 5 |- (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A}))) = (F |` (C \ {A}))
21	5, 6, 20	3eqtri 1912	. . . 4 |- (G |` (C \ {A})) = (F |` (C \ {A}))
22	21	fveq1i 4682	. . 3 |- ((G |` (C \ {A}))` D) = ((F |` (C \ {A}))` D)
23	2, 22	syl5eq 1940	. 2 |- (D e. (C \ {A}) -> ((G |` (C \ {A}))` D) = (F` D))
24	1, 23	eqtr3d 1927	1 |- (D e. (C \ {A}) -> (G` D) = (F` D))

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  _Vcvv 2292   \ cdif 2590   u. cun 2591   i^i cin 2592  (/)c0 2875  {csn 3044  <.cop 3046   |` cres 3988   Fn wfn 3993  -1-1-onto->wf1o 3997  ` cfv 3998

This theorem is referenced by:  facnn 8185  acdc2lem2 8758  acdc5lem2 8761  ruclem8 8786

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014

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