Theorem fvsnun2 3787

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. See also fvsnun1 3786.

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 3725	. 2 |- (D e. (C \ {A}) -> ((G |` (C \ {A}))` D) = (G` D))
2		fvres 3725	. . 3 |- (D e. (C \ {A}) -> ((F |` (C \ {A}))` D) = (F` D))
3		fvsnun.3	. . . . . 6 |- G = ({<.A, B>.} u. (F |` (C \ {A})))
4		reseq1 3360	. . . . . 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 3371	. . . . 5 |- (({<.A, B>.} u. (F |` (C \ {A}))) |` (C \ {A})) = (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A})))
7		difdisj 2333	. . . . . . . 8 |- ({A} i^i (C \ {A})) = (/)
8		fvsnun.1	. . . . . . . . . . 11 |- A e. V
9		fvsnun.2	. . . . . . . . . . 11 |- B e. V
10	8, 9	f1osn 3710	. . . . . . . . . 10 |- {<.A, B>.}:{A}-1-1-onto->{B}
11		f1ofn 3681	. . . . . . . . . 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 3589	. . . . . . . . 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 3382	. . . . . . 7 |- ((F |` (C \ {A})) |` (C \ {A})) = (F |` (C \ {A}))
17	15, 16	uneq12i 2178	. . . . . 6 |- (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A}))) = ((/) u. (F |` (C \ {A})))
18		uncom 2172	. . . . . 6 |- ((/) u. (F |` (C \ {A}))) = ((F |` (C \ {A})) u. (/))
19		un0 2293	. . . . . 6 |- ((F |` (C \ {A})) u. (/)) = (F |` (C \ {A}))
20	17, 18, 19	3eqtr 1496	. . . . 5 |- (({<.A, B>.} |` (C \ {A})) u. ((F |` (C \ {A})) |` (C \ {A}))) = (F |` (C \ {A}))
21	5, 6, 20	3eqtr 1496	. . . 4 |- (G |` (C \ {A})) = (F |` (C \ {A}))
22	21	fveq1i 3716	. . 3 |- ((G |` (C \ {A}))` D) = ((F |` (C \ {A}))` D)
23	2, 22	syl5eq 1516	. 2 |- (D e. (C \ {A}) -> ((G |` (C \ {A}))` D) = (F` D))
24	1, 23	eqtr3d 1506	1 |- (D e. (C \ {A}) -> (G` D) = (F` D))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 954   e. wcel 956  Vcvv 1807   \ cdif 2040   u. cun 2041   i^i cin 2042  (/)c0 2276  {csn 2405  <.cop 2407   |` cres 3167   Fn wfn 3172  -1-1-onto->wf1o 3176  ` cfv 3177

This theorem is referenced by:  facnnt 6878  acdc2lem2 7439  acdc5lem2 7442  ruclem8 7468

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193

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