Theorem coi1 5095

Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by set.mm contributors, 22-Apr-2004.) (Revised by Scott Fenton, 14-Apr-2021.)

Assertion

Ref	Expression
coi1	|- (A o. _I ) = A

Proof of Theorem coi1

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		brco 4884	. . 3 |- (x(A o. _I )y <-> E.z(x _I z /\ zAy))
2		vex 2863	. . . . . . 7 |- z e. _V
3	2	ideq 4871	. . . . . 6 |- (x _I z <-> x = z)
4		equcom 1680	. . . . . 6 |- (x = z <-> z = x)
5	3, 4	bitri 240	. . . . 5 |- (x _I z <-> z = x)
6	5	anbi1i 676	. . . 4 |- ((x _I z /\ zAy) <-> (z = x /\ zAy))
7	6	exbii 1582	. . 3 |- (E.z(x _I z /\ zAy) <-> E.z(z = x /\ zAy))
8		vex 2863	. . . 4 |- x e. _V
9		breq1 4643	. . . 4 |- (z = x -> (zAy <-> xAy))
10	8, 9	ceqsexv 2895	. . 3 |- (E.z(z = x /\ zAy) <-> xAy)
11	1, 7, 10	3bitri 262	. 2 |- (x(A o. _I )y <-> xAy)
12	11	eqbrriv 4852	1 |- (A o. _I ) = A

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   class class class wbr 4640   o. ccom 4722   _I cid 4764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-id 4768

This theorem is referenced by:  coi2  5096  coires1  5097  funi  5138  fcoi1  5241