Theorem coi2 3497

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

Assertion

Ref	Expression
coi2	|- (Rel A -> (I o. A) = A)

Proof of Theorem coi2

Step	Hyp	Ref	Expression
1		dfrel2 3471	. . . 4 |- (Rel A <-> `'`'A = A)
2		cnvi 3433	. . . . 5 |- `'I = I
3		coeq2 3271	. . . . . 6 |- (`'`'A = A -> (`'I o. `'`'A) = (`'I o. A))
4		coeq1 3270	. . . . . 6 |- (`'I = I -> (`'I o. A) = (I o. A))
5	3, 4	sylan9eq 1519	. . . . 5 |- ((`'`'A = A /\ `'I = I) -> (`'I o. `'`'A) = (I o. A))
6	2, 5	mpan2 694	. . . 4 |- (`'`'A = A -> (`'I o. `'`'A) = (I o. A))
7	1, 6	sylbi 199	. . 3 |- (Rel A -> (`'I o. `'`'A) = (I o. A))
8		cnvco 3289	. . . 4 |- `'(`'A o. I) = (`'I o. `'`'A)
9		relcnv 3419	. . . . . 6 |- Rel `'A
10		coi1 3496	. . . . . 6 |- (Rel `'A -> (`'A o. I) = `'A)
11	9, 10	ax-mp 7	. . . . 5 |- (`'A o. I) = `'A
12		cnveq 3281	. . . . 5 |- ((`'A o. I) = `'A -> `'(`'A o. I) = `'`'A)
13	11, 12	ax-mp 7	. . . 4 |- `'(`'A o. I) = `'`'A
14	8, 13	eqtr3 1489	. . 3 |- (`'I o. `'`'A) = `'`'A
15	7, 14	syl5reqr 1514	. 2 |- (Rel A -> (I o. A) = `'`'A)
16	1	biimp 151	. 2 |- (Rel A -> `'`'A = A)
17	15, 16	eqtrd 1499	1 |- (Rel A -> (I o. A) = A)

Syntax hints:   -> wi 3   = wceq 953  Icid 2820  `'ccnv 3159   o. ccom 3164  Rel wrel 3165

This theorem is referenced by:  funi 3531

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177

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