Theorem fvsnun2 3796

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 3795.

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 3734	. 2 |- (D e. (C \ {A}) -> ((G |` (C \ {A}))` D) = (G` D))
2		fvres 3734	. . 3 |- (D e. (C \ {A}) -> ((F |` (C \ {A}))` D) = (F` D))
3		fvsnun.3	. . . . . 6 |- G = ({<.A, B>.} u. (F |` (C \ {A})))
4		reseq1 3368	. . . . . 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 3379	. . . . 5 |- (({<.A, B>.} u. (F |` (C \ {A}))) |` (C \ {A})) = (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A})))
7		difdisj 2337	. . . . . . . 8 |- ({A} i^i (C \ {A})) = (/)
8		fvsnun.1	. . . . . . . . . . 11 |- A e. V
9		fvsnun.2	. . . . . . . . . . 11 |- B e. V
10	8, 9	f1osn 3719	. . . . . . . . . 10 |- {<.A, B>.}:{A}-1-1-onto->{B}
11		f1ofn 3690	. . . . . . . . . 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 3597	. . . . . . . . 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 189	. . . . . . 7 |- ({<.A, B>.} |` (C \ {A})) = (/)
16		residm 3390	. . . . . . 7 |- ((F |` (C \ {A})) |` (C \ {A})) = (F |` (C \ {A}))
17	15, 16	uneq12i 2182	. . . . . 6 |- (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A}))) = ((/) u. (F |` (C \ {A})))
18		uncom 2176	. . . . . 6 |- ((/) u. (F |` (C \ {A}))) = ((F |` (C \ {A})) u. (/))
19		un0 2297	. . . . . 6 |- ((F |` (C \ {A})) u. (/)) = (F |` (C \ {A}))
20	17, 18, 19	3eqtr 1499	. . . . 5 |- (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A}))) = (F |` (C \ {A}))
21	5, 6, 20	3eqtr 1499	. . . 4 |- (G |` (C \ {A})) = (F |` (C \ {A}))
22	21	fveq1i 3725	. . 3 |- ((G |` (C \ {A}))` D) = ((F |` (C \ {A}))` D)
23	2, 22	syl5eq 1519	. 2 |- (D e. (C \ {A}) -> ((G |` (C \ {A}))` D) = (F` D))
24	1, 23	eqtr3d 1509	1 |- (D e. (C \ {A}) -> (G` D) = (F` D))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 956   e. wcel 958  Vcvv 1811   \ cdif 2044   u. cun 2045   i^i cin 2046  (/)c0 2280  {csn 2409  <.cop 2411   |` cres 3172   Fn wfn 3177  -1-1-onto->wf1o 3181  ` cfv 3182

This theorem is referenced by:  facnnt 6933  acdc2lem2 7489  acdc5lem2 7492  ruclem8 7517

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198

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