Theorem coi2 3503

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 3477	. . . 4 |- (Rel A <-> `'`'A = A)
2		cnvi 3439	. . . . 5 |- `'I = I
3		coeq2 3277	. . . . . 6 |- (`'`'A = A -> (`'I o. `'`'A) = (`'I o. A))
4		coeq1 3276	. . . . . 6 |- (`'I = I -> (`'I o. A) = (I o. A))
5	3, 4	sylan9eq 1524	. . . . 5 |- ((`'`'A = A /\ `'I = I) -> (`'I o. `'`'A) = (I o. A))
6	2, 5	mpan2 695	. . . 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 3295	. . . 4 |- `'(`'A o. I) = (`'I o. `'`'A)
9		relcnv 3427	. . . . . 6 |- Rel `'A
10		coi1 3502	. . . . . 6 |- (Rel `'A -> (`'A o. I) = `'A)
11	9, 10	ax-mp 7	. . . . 5 |- (`'A o. I) = `'A
12		cnveq 3287	. . . . 5 |- ((`'A o. I) = `'A -> `'(`'A o. I) = `'`'A)
13	11, 12	ax-mp 7	. . . 4 |- `'(`'A o. I) = `'`'A
14	8, 13	eqtr3 1494	. . 3 |- (`'I o. `'`'A) = `'`'A
15	7, 14	syl5reqr 1519	. 2 |- (Rel A -> (I o. A) = `'`'A)
16	1	biimp 151	. 2 |- (Rel A -> `'`'A = A)
17	15, 16	eqtrd 1504	1 |- (Rel A -> (I o. A) = A)

Syntax hints:   -> wi 3   = wceq 954  Icid 2826  `'ccnv 3164   o. ccom 3169  Rel wrel 3170

This theorem is referenced by:  funi 3537

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182

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