Theorem fvsnun1 3795

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 3796.

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
fvsnun1	|- (G` A) = B

Proof of Theorem fvsnun1

Step	Hyp	Ref	Expression
1		fvsnun.1	. . . 4 |- A e. V
2	1	snid 2435	. . 3 |- A e. {A}
3		fvres 3734	. . 3 |- (A e. {A} -> ((G |` {A})` A) = (G` A))
4	2, 3	ax-mp 7	. 2 |- ((G |` {A})` A) = (G` A)
5		fvsnun.3	. . . . . 6 |- G = ({<.A, B>.} u. (F |` (C \ {A})))
6		reseq1 3368	. . . . . 6 |- (G = ({<.A, B>.} u. (F |` (C \ {A}))) -> (G |` {A}) = (({<.A, B>.} u. (F |` (C \ {A}))) |` {A}))
7	5, 6	ax-mp 7	. . . . 5 |- (G |` {A}) = (({<.A, B>.} u. (F |` (C \ {A}))) |` {A})
8		resundir 3379	. . . . 5 |- (({<.A, B>.} u. (F |` (C \ {A}))) |` {A}) = (({<.A, B>.} |` {A}) u. ((F |` (C \ {A})) |` {A}))
9		incom 2208	. . . . . . . . 9 |- ((C \ {A}) i^i {A}) = ({A} i^i (C \ {A}))
10		difdisj 2337	. . . . . . . . 9 |- ({A} i^i (C \ {A})) = (/)
11	9, 10	eqtr 1495	. . . . . . . 8 |- ((C \ {A}) i^i {A}) = (/)
12		resdisj 3471	. . . . . . . 8 |- (((C \ {A}) i^i {A}) = (/) -> ((F |` (C \ {A})) |` {A}) = (/))
13	11, 12	ax-mp 7	. . . . . . 7 |- ((F |` (C \ {A})) |` {A}) = (/)
14	13	uneq2i 2181	. . . . . 6 |- (({<.A, B>.} |` {A}) u. ((F |` (C \ {A})) |` {A})) = (({<.A, B>.} |` {A}) u. (/))
15		un0 2297	. . . . . 6 |- (({<.A, B>.} |` {A}) u. (/)) = ({<.A, B>.} |` {A})
16	14, 15	eqtr 1495	. . . . 5 |- (({<.A, B>.} |` {A}) u. ((F |` (C \ {A})) |` {A})) = ({<.A, B>.} |` {A})
17	7, 8, 16	3eqtr 1499	. . . 4 |- (G |` {A}) = ({<.A, B>.} |` {A})
18	17	fveq1i 3725	. . 3 |- ((G |` {A})` A) = (({<.A, B>.} |` {A})` A)
19		fvres 3734	. . . 4 |- (A e. {A} -> (({<.A, B>.} |` {A})` A) = ({<.A, B>.}` A))
20	2, 19	ax-mp 7	. . 3 |- (({<.A, B>.} |` {A})` A) = ({<.A, B>.}` A)
21		fvsnun.2	. . . 4 |- B e. V
22	1, 21	fvsn 3794	. . 3 |- ({<.A, B>.}` A) = B
23	18, 20, 22	3eqtr 1499	. 2 |- ((G |` {A})` A) = B
24	4, 23	eqtr3 1497	1 |- (G` A) = B

Syntax hints:   = wceq 956   e. wcel 958  Vcvv 1811   \ cdif 2044   u. cun 2045   i^i cin 2046  (/)c0 2280  {csn 2409  <.cop 2411   |` cres 3172  ` cfv 3182

This theorem is referenced by:  fac0 6934  acdc2lem2 7489  acdc5lem2 7492  ruclem7 7516

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-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  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-fv 3198

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