Theorem fvsnun1 3801

Description: The value of a function with one of its ordered pairs replaced, at the replaced ordered pair. See also fvsnun2 3802.

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 2439	. . 3 |- A e. {A}
3		fvres 3740	. . 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 3374	. . . . . 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 3385	. . . . 5 |- (({<.A, B>.} u. (F |` (C \ {A}))) |` {A}) = (({<.A, B>.} |` {A}) u. ((F |` (C \ {A})) |` {A}))
9		incom 2211	. . . . . . . . 9 |- ((C \ {A}) i^i {A}) = ({A} i^i (C \ {A}))
10		difdisj 2341	. . . . . . . . 9 |- ({A} i^i (C \ {A})) = (/)
11	9, 10	eqtr 1498	. . . . . . . 8 |- ((C \ {A}) i^i {A}) = (/)
12		resdisj 3477	. . . . . . . 8 |- (((C \ {A}) i^i {A}) = (/) -> ((F |` (C \ {A})) |` {A}) = (/))
13	11, 12	ax-mp 7	. . . . . . 7 |- ((F |` (C \ {A})) |` {A}) = (/)
14	13	uneq2i 2184	. . . . . 6 |- (({<.A, B>.} |` {A}) u. ((F |` (C \ {A})) |` {A})) = (({<.A, B>.} |` {A}) u. (/))
15		un0 2301	. . . . . 6 |- (({<.A, B>.} |` {A}) u. (/)) = ({<.A, B>.} |` {A})
16	14, 15	eqtr 1498	. . . . 5 |- (({<.A, B>.} |` {A}) u. ((F |` (C \ {A})) |` {A})) = ({<.A, B>.} |` {A})
17	7, 8, 16	3eqtr 1502	. . . 4 |- (G |` {A}) = ({<.A, B>.} |` {A})
18	17	fveq1i 3731	. . 3 |- ((G |` {A})` A) = (({<.A, B>.} |` {A})` A)
19		fvres 3740	. . . 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 3800	. . 3 |- ({<.A, B>.}` A) = B
23	18, 20, 22	3eqtr 1502	. 2 |- ((G |` {A})` A) = B
24	4, 23	eqtr3 1500	1 |- (G` A) = B

Syntax hints:   = wceq 958   e. wcel 960  Vcvv 1814   \ cdif 2047   u. cun 2048   i^i cin 2049  (/)c0 2283  {csn 2413  <.cop 2415   |` cres 3178  ` cfv 3188

This theorem is referenced by:  fac0 6934  acdc2lem2 7490  acdc5lem2 7493  ruclem7 7517

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204

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