Theorem coi1 3516

Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36.

Assertion

Ref	Expression
coi1	|- (Rel A -> (A o. I) = A)

Proof of Theorem coi1

Step	Hyp	Ref	Expression
1		relco 3490	. 2 |- Rel (A o. I)
2		visset 1816	. . . . . . 7 |- x e. V
3		visset 1816	. . . . . . 7 |- y e. V
4	2, 3	opelco 3294	. . . . . 6 |- (<.x, y>. e. (A o. I) <-> E.z(xIz /\ zAy))
5		visset 1816	. . . . . . . . . 10 |- z e. V
6	5	ideq 3283	. . . . . . . . 9 |- (xIz <-> x = z)
7		eqcom 1480	. . . . . . . . 9 |- (x = z <-> z = x)
8	6, 7	bitr 173	. . . . . . . 8 |- (xIz <-> z = x)
9	8	anbi1i 483	. . . . . . 7 |- ((xIz /\ zAy) <-> (z = x /\ zAy))
10	9	exbii 1053	. . . . . 6 |- (E.z(xIz /\ zAy) <-> E.z(z = x /\ zAy))
11		breq1 2627	. . . . . . 7 |- (z = x -> (zAy <-> xAy))
12	2, 11	ceqsexv 1838	. . . . . 6 |- (E.z(z = x /\ zAy) <-> xAy)
13	4, 10, 12	3bitr 177	. . . . 5 |- (<.x, y>. e. (A o. I) <-> xAy)
14		df-br 2625	. . . . 5 |- (xAy <-> <.x, y>. e. A)
15	13, 14	bitr 173	. . . 4 |- (<.x, y>. e. (A o. I) <-> <.x, y>. e. A)
16	15	gen2 985	. . 3 |- A.xA.y(<.x, y>. e. (A o. I) <-> <.x, y>. e. A)
17		eqrel 3256	. . 3 |- ((Rel (A o. I) /\ Rel A) -> ((A o. I) = A <-> A.xA.y(<.x, y>. e. (A o. I) <-> <.x, y>. e. A)))
18	16, 17	mpbiri 194	. 2 |- ((Rel (A o. I) /\ Rel A) -> (A o. I) = A)
19	1, 18	mpan 697	1 |- (Rel A -> (A o. I) = A)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  <.cop 2415   class class class wbr 2624  Icid 2837   o. ccom 3180  Rel wrel 3181

This theorem is referenced by:  coi2 3517

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-co 3193

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