Theorem fvres 2840

Description: The value of a restricted function.

Assertion

Ref	Expression
fvres	|- (A e. B -> ((F |` B)` A) = (F` A))

Proof of Theorem fvres

Step	Hyp	Ref	Expression
1		snssi 1851	. . . . . . 7 |- (A e. B -> {A} (_ B)
2		resabs1 2592	. . . . . . 7 |- ({A} (_ B -> ((F |` B) |` {A}) = (F |` {A}))
3		rneq 2555	. . . . . . 7 |- (((F |` B) |` {A}) = (F |` {A}) -> ran ((F |` B) |` {A}) = ran (F |` {A}))
4	1, 2, 3	3syl 21	. . . . . 6 |- (A e. B -> ran ((F |` B) |` {A}) = ran (F |` {A}))
5		df-ima 2431	. . . . . 6 |- ((F |` B)"{A}) = ran ((F |` B) |` {A})
6		df-ima 2431	. . . . . 6 |- (F"{A}) = ran (F |` {A})
7	4, 5, 6	3eqtr4g 1147	. . . . 5 |- (A e. B -> ((F |` B)"{A}) = (F"{A}))
8	7	cleq1d 1109	. . . 4 |- (A e. B -> (((F |` B)"{A}) = {x} <-> (F"{A}) = {x}))
9	8	biabdv 1183	. . 3 |- (A e. B -> {x | ((F |` B)"{A}) = {x}} = {x | (F"{A}) = {x}})
10	9	unieqd 1929	. 2 |- (A e. B -> U.{x | ((F |` B)"{A}) = {x}} = U.{x | (F"{A}) = {x}})
11		df-fv 2438	. 2 |- ((F |` B)` A) = U.{x | ((F |` B)"{A}) = {x}}
12		df-fv 2438	. 2 |- (F` A) = U.{x | (F"{A}) = {x}}
13	10, 11, 12	3eqtr4g 1147	1 |- (A e. B -> ((F |` B)` A) = (F` A))

Syntax hints:   -> wi 2  {cab 1090   = wceq 1091   e. wcel 1092   (_ wss 1487  {csn 1808  U.cuni 1919  ran crn 2411   |` cres 2412  "cima 2413  ` cfv 2422

This theorem is referenced by:  funssfv 2841  fveqres 2851  fvreseq 2882  fnressn 2897  fressnfv 2898  fvresi 2901  funfvima 2904  abrexexlem1 2910  isoid 2933  f1oweOLD 2944  tfrlem5 2953  tz7.44-2 2967  frzer 2990  frsuc 2991  tz7.48lem 2993  tz7.48-2 2995  df1st2 3098  addpiord 3806  mulpiord 3807  seqrn 4673  facnnt 4870  fac0 4871  fac1 4872  facp1t 4873  ruclem7 4891  ruclem8 4892

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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