Theorem iss 3403

Description: A subclass of the identity function is the identity function restricted to its domain.

Assertion

Ref	Expression
iss	|- (A (_ I <-> A = (I |` dom A))

Proof of Theorem iss

Step	Hyp	Ref	Expression
1		ssel 2066	. . . . . . . 8 |- (A (_ I -> (<.x, y>. e. A -> <.x, y>. e. I))
2		df-br 2625	. . . . . . . . 9 |- (xIy <-> <.x, y>. e. I)
3		visset 1816	. . . . . . . . . 10 |- y e. V
4	3	ideq 3283	. . . . . . . . 9 |- (xIy <-> x = y)
5	2, 4	bitr3 175	. . . . . . . 8 |- (<.x, y>. e. I <-> x = y)
6	1, 5	syl6ib 212	. . . . . . 7 |- (A (_ I -> (<.x, y>. e. A -> x = y))
7	6	pm4.71rd 641	. . . . . 6 |- (A (_ I -> (<.x, y>. e. A <-> (x = y /\ <.x, y>. e. A)))
8		eqcom 1480	. . . . . . . . . . . . 13 |- (x = y <-> y = x)
9	8	anbi1i 483	. . . . . . . . . . . 12 |- ((x = y /\ <.x, y>. e. A) <-> (y = x /\ <.x, y>. e. A))
10	7, 9	syl6bb 538	. . . . . . . . . . 11 |- (A (_ I -> (<.x, y>. e. A <-> (y = x /\ <.x, y>. e. A)))
11	10	exbidv 1281	. . . . . . . . . 10 |- (A (_ I -> (E.y<.x, y>. e. A <-> E.y(y = x /\ <.x, y>. e. A)))
12		visset 1816	. . . . . . . . . . 11 |- x e. V
13		opeq2 2492	. . . . . . . . . . . 12 |- (y = x -> <.x, y>. = <.x, x>.)
14	13	eleq1d 1543	. . . . . . . . . . 11 |- (y = x -> (<.x, y>. e. A <-> <.x, x>. e. A))
15	12, 14	ceqsexv 1838	. . . . . . . . . 10 |- (E.y(y = x /\ <.x, y>. e. A) <-> <.x, x>. e. A)
16	11, 15	syl6bb 538	. . . . . . . . 9 |- (A (_ I -> (E.y<.x, y>. e. A <-> <.x, x>. e. A))
17	12	eldm2 3314	. . . . . . . . 9 |- (x e. dom A <-> E.y<.x, y>. e. A)
18	16, 17	syl5bb 534	. . . . . . . 8 |- (A (_ I -> (x e. dom A <-> <.x, x>. e. A))
19	18	anbi2d 618	. . . . . . 7 |- (A (_ I -> ((x = y /\ x e. dom A) <-> (x = y /\ <.x, x>. e. A)))
20		opeq2 2492	. . . . . . . . 9 |- (x = y -> <.x, x>. = <.x, y>.)
21	20	eleq1d 1543	. . . . . . . 8 |- (x = y -> (<.x, x>. e. A <-> <.x, y>. e. A))
22	21	pm5.32i 647	. . . . . . 7 |- ((x = y /\ <.x, x>. e. A) <-> (x = y /\ <.x, y>. e. A))
23	19, 22	syl6bb 538	. . . . . 6 |- (A (_ I -> ((x = y /\ x e. dom A) <-> (x = y /\ <.x, y>. e. A)))
24	7, 23	bitr4d 533	. . . . 5 |- (A (_ I -> (<.x, y>. e. A <-> (x = y /\ x e. dom A)))
25	3	opelres 3378	. . . . . 6 |- (<.x, y>. e. (I |` dom A) <-> (<.x, y>. e. I /\ x e. dom A))
26	5	anbi1i 483	. . . . . 6 |- ((<.x, y>. e. I /\ x e. dom A) <-> (x = y /\ x e. dom A))
27	25, 26	bitr2 174	. . . . 5 |- ((x = y /\ x e. dom A) <-> <.x, y>. e. (I |` dom A))
28	24, 27	syl6bb 538	. . . 4 |- (A (_ I -> (<.x, y>. e. A <-> <.x, y>. e. (I |` dom A)))
29	28	19.21aivv 1289	. . 3 |- (A (_ I -> A.xA.y(<.x, y>. e. A <-> <.x, y>. e. (I |` dom A)))
30		reli 3279	. . . . 5 |- Rel I
31		relss 3252	. . . . 5 |- (A (_ I -> (Rel I -> Rel A))
32	30, 31	mpi 44	. . . 4 |- (A (_ I -> Rel A)
33		relres 3393	. . . . 5 |- Rel (I |` dom A)
34		eqrel 3256	. . . . 5 |- ((Rel A /\ Rel (I |` dom A)) -> (A = (I |` dom A) <-> A.xA.y(<.x, y>. e. A <-> <.x, y>. e. (I |` dom A))))
35	33, 34	mpan2 698	. . . 4 |- (Rel A -> (A = (I |` dom A) <-> A.xA.y(<.x, y>. e. A <-> <.x, y>. e. (I |` dom A))))
36	32, 35	syl 10	. . 3 |- (A (_ I -> (A = (I |` dom A) <-> A.xA.y(<.x, y>. e. A <-> <.x, y>. e. (I |` dom A))))
37	29, 36	mpbird 196	. 2 |- (A (_ I -> A = (I |` dom A))
38		resss 3389	. . 3 |- (I |` dom A) (_ I
39		sseq1 2085	. . 3 |- (A = (I |` dom A) -> (A (_ I <-> (I |` dom A) (_ I))
40	38, 39	mpbiri 194	. 2 |- (A = (I |` dom A) -> A (_ I)
41	37, 40	impbi 157	1 |- (A (_ I <-> A = (I |` dom A))

Syntax hints:   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982   (_ wss 2050  <.cop 2415   class class class wbr 2624  Icid 2837  dom cdm 3176   |` cres 3178  Rel wrel 3181

This theorem is referenced by:  f1ococnv2 3714

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-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-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-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-dm 3194  df-res 3196

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