Theorem fvi 2900

Description: The value of the identity function.

Assertion

Ref	Expression
fvi	|- (A e. B -> (I` A) = A)

Proof of Theorem fvi

Step	Hyp	Ref	Expression
1		fveq2 2832	. . 3 |- (x = A -> (I` x) = (I` A))
2		id 9	. . 3 |- (x = A -> x = A)
3	1, 2	cleq12d 1115	. 2 |- (x = A -> ((I` x) = x <-> (I` A) = A))
4		funi 2692	. . . . 5 |- Fun I
5		dmi 2545	. . . . 5 |- dom I = V
6	4, 5	pm3.2i 234	. . . 4 |- (Fun I /\ dom I = V)
7		df-fn 2433	. . . 4 |- (I Fn V <-> (Fun I /\ dom I = V))
8	6, 7	mpbir 165	. . 3 |- I Fn V
9		visset 1350	. . 3 |- x e. V
10		cleqid 1102	. . . . . 6 |- x = x
11	9, 9	ideq 2127	. . . . . 6 |- (xIx <-> x = x)
12	10, 11	mpbir 165	. . . . 5 |- xIx
13		df-br 2063	. . . . 5 |- (xIx <-> <.x, x>. e. I)
14	12, 13	mpbi 164	. . . 4 |- <.x, x>. e. I
15	9	fnfvop 2856	. . . 4 |- ((I Fn V /\ x e. V) -> ((I` x) = x <-> <.x, x>. e. I))
16	14, 15	mpbiri 169	. . 3 |- ((I Fn V /\ x e. V) -> (I` x) = x)
17	8, 9, 16	mp2an 520	. 2 |- (I` x) = x
18	3, 17	vtoclg 1383	1 |- (A e. B -> (I` A) = A)

Syntax hints:   -> wi 2   /\ wa 196   = weq 797   = wceq 1091   e. wcel 1092  Vcvv 1348  <.cop 1810   class class class wbr 2054  Icid 2057  dom cdm 2410  Fun wfun 2416   Fn wfn 2417  ` cfv 2422

This theorem is referenced by:  fvresi 2901  isoid 2933  fac1 4872  facp1t 4873

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-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  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-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438

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