Theorem cores2 4410

Description: Absorption of a reverse (preimage) restriction of the second member of a class composition.

Assertion

Ref	Expression
cores2	|- (dom A C_ C -> (A o. `'(`'B |` C)) = (A o. B))

Proof of Theorem cores2

Step	Hyp	Ref	Expression
1		dfdm4 4151	. . . . . 6 |- dom A = ran `' A
2	1	sseq1i 2641	. . . . 5 |- (dom A C_ C <-> ran `' A C_ C)
3		cores 4400	. . . . 5 |- (ran `' A C_ C -> ((`'B |` C) o. `'A) = (`'B o. `'A))
4	2, 3	sylbi 216	. . . 4 |- (dom A C_ C -> ((`'B |` C) o. `'A) = (`'B o. `'A))
5		cnvco 4145	. . . . 5 |- `'(A o. `'(`'B |` C)) = (`'`'(`'B |` C) o. `'A)
6		relres 4242	. . . . . . 7 |- Rel (`'B |` C)
7		dfrel2 4358	. . . . . . 7 |- (Rel (`'B |` C) <-> `'`'(`'B |` C) = (`'B |` C))
8	6, 7	mpbi 206	. . . . . 6 |- `'`'(`'B |` C) = (`'B |` C)
9	8	coeq1i 4125	. . . . 5 |- (`'`'(`'B |` C) o. `'A) = ((`'B |` C) o. `'A)
10	5, 9	eqtri 1908	. . . 4 |- `'(A o. `'(`'B |` C)) = ((`'B |` C) o. `'A)
11		cnvco 4145	. . . 4 |- `'(A o. B) = (`'B o. `'A)
12	4, 10, 11	3eqtr4g 1953	. . 3 |- (dom A C_ C -> `'(A o. `'(`'B |` C)) = `'(A o. B))
13		cnveq 4135	. . 3 |- (`'(A o. `'(`'B |` C)) = `'(A o. B) -> `'`'(A o. `'(`'B |` C)) = `'`'(A o. B))
14	12, 13	syl 12	. 2 |- (dom A C_ C -> `'`'(A o. `'(`'B |` C)) = `'`'(A o. B))
15		relco 4392	. . 3 |- Rel (A o. `'(`'B |` C))
16		dfrel2 4358	. . 3 |- (Rel (A o. `'(`'B |` C)) <-> `'`'(A o. `'(`'B |` C)) = (A o. `'(`'B |` C)))
17	15, 16	mpbi 206	. 2 |- `'`'(A o. `'(`'B |` C)) = (A o. `'(`'B |` C))
18		relco 4392	. . 3 |- Rel (A o. B)
19		dfrel2 4358	. . 3 |- (Rel (A o. B) <-> `'`'(A o. B) = (A o. B))
20	18, 19	mpbi 206	. 2 |- `'`'(A o. B) = (A o. B)
21	14, 17, 20	3eqtr3g 1952	1 |- (dom A C_ C -> (A o. `'(`'B |` C)) = (A o. B))

Syntax hints:   -> wi 3   = wceq 1298   C_ wss 2593  `'ccnv 3985  dom cdm 3986  ran crn 3987   |` cres 3988   o. ccom 3990  Rel wrel 3991

This theorem is referenced by:  fcoi1 4584

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006

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