Theorem coi1 5094

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 4883	. . 3 |- (x(A o. _I )y <-> E.z(x _I z /\ zAy))
2		vex 2862	. . . . . . 7 |- z e. _V
3	2	ideq 4870	. . . . . 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 2862	. . . 4 |- x e. _V
9		breq1 4642	. . . 4 |- (z = x -> (zAy <-> xAy))
10	8, 9	ceqsexv 2894	. . 3 |- (E.z(z = x /\ zAy) <-> xAy)
11	1, 7, 10	3bitri 262	. 2 |- (x(A o. _I )y <-> xAy)
12	11	eqbrriv 4851	1 |- (A o. _I ) = A

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   class class class wbr 4639   o. ccom 4721   _I cid 4763

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-id 4767

This theorem is referenced by:  coi2  5095  coires1  5096  funi  5137  fcoi1  5240