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Theorem coi1 4779
 Description: Composition with the identity relation. Part of Theorem 3.7(i) of [Monk1] p. 36. (Contributed by NM, 22-Apr-2004.)
Assertion
Ref Expression
coi1 (Rel A → (A ∘ I ) = A)

Proof of Theorem coi1
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 4762 . 2 Rel (A ∘ I )
2 vex 2554 . . . . . 6 x V
3 vex 2554 . . . . . 6 y V
42, 3opelco 4450 . . . . 5 (⟨x, y (A ∘ I ) ↔ z(x I z zAy))
5 vex 2554 . . . . . . . . . 10 z V
65ideq 4431 . . . . . . . . 9 (x I zx = z)
7 equcom 1590 . . . . . . . . 9 (x = zz = x)
86, 7bitri 173 . . . . . . . 8 (x I zz = x)
98anbi1i 431 . . . . . . 7 ((x I z zAy) ↔ (z = x zAy))
109exbii 1493 . . . . . 6 (z(x I z zAy) ↔ z(z = x zAy))
11 breq1 3758 . . . . . . 7 (z = x → (zAyxAy))
122, 11ceqsexv 2587 . . . . . 6 (z(z = x zAy) ↔ xAy)
1310, 12bitri 173 . . . . 5 (z(x I z zAy) ↔ xAy)
144, 13bitri 173 . . . 4 (⟨x, y (A ∘ I ) ↔ xAy)
15 df-br 3756 . . . 4 (xAy ↔ ⟨x, y A)
1614, 15bitri 173 . . 3 (⟨x, y (A ∘ I ) ↔ ⟨x, y A)
1716eqrelriv 4376 . 2 ((Rel (A ∘ I ) Rel A) → (A ∘ I ) = A)
181, 17mpan 400 1 (Rel A → (A ∘ I ) = A)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ⟨cop 3370   class class class wbr 3755   I cid 4016   ∘ ccom 4292  Rel wrel 4293 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-co 4297 This theorem is referenced by:  coi2  4780  coires1  4781  relcoi1  4792  fcoi1  5013
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