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Theorem coi2 4780
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
coi2 (Rel A → ( I ∘ A) = A)

Proof of Theorem coi2
StepHypRef Expression
1 cnvco 4463 . . 3 (A ∘ I ) = ( I ∘ A)
2 relcnv 4646 . . . . 5 Rel A
3 coi1 4779 . . . . 5 (Rel A → (A ∘ I ) = A)
42, 3ax-mp 7 . . . 4 (A ∘ I ) = A
54cnveqi 4453 . . 3 (A ∘ I ) = A
61, 5eqtr3i 2059 . 2 ( I ∘ A) = A
7 dfrel2 4714 . . 3 (Rel AA = A)
8 cnvi 4671 . . . 4 I = I
9 coeq2 4437 . . . . 5 (A = A → ( I ∘ A) = ( I ∘ A))
10 coeq1 4436 . . . . 5 ( I = I → ( I ∘ A) = ( I ∘ A))
119, 10sylan9eq 2089 . . . 4 ((A = A I = I ) → ( I ∘ A) = ( I ∘ A))
128, 11mpan2 401 . . 3 (A = A → ( I ∘ A) = ( I ∘ A))
137, 12sylbi 114 . 2 (Rel A → ( I ∘ A) = ( I ∘ A))
147biimpi 113 . 2 (Rel AA = A)
156, 13, 143eqtr3a 2093 1 (Rel A → ( I ∘ A) = A)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   I cid 4016  ccnv 4287  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-bnd 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-cnv 4296  df-co 4297
This theorem is referenced by:  relcoi2  4791  funi  4875  fcoi2  5014
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