Theorem cores2 3507

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

Assertion

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

Proof of Theorem cores2

Step	Hyp	Ref	Expression
1		dfdm4 3305	. . . . . 6 |- dom A = ran `' A
2	1	sseq1i 2085	. . . . 5 |- (dom A (_ C <-> ran `' A (_ C)
3		cores 3499	. . . . 5 |- (ran `' A (_ C -> ((`'B |` C) o. `'A) = (`'B o. `'A))
4	2, 3	sylbi 199	. . . 4 |- (dom A (_ C -> ((`'B |` C) o. `'A) = (`'B o. `'A))
5		cnvco 3300	. . . . 5 |- `'(A o. `'(`'B |` C)) = (`'`'(`'B |` C) o. `'A)
6		relres 3387	. . . . . . 7 |- Rel (`'B |` C)
7		dfrel2 3485	. . . . . . 7 |- (Rel (`'B |` C) <-> `'`'(`'B |` C) = (`'B |` C))
8	6, 7	mpbi 189	. . . . . 6 |- `'`'(`'B |` C) = (`'B |` C)
9	8	coeq1i 3283	. . . . 5 |- (`'`'(`'B |` C) o. `'A) = ((`'B |` C) o. `'A)
10	5, 9	eqtr 1495	. . . 4 |- `'(A o. `'(`'B |` C)) = ((`'B |` C) o. `'A)
11		cnvco 3300	. . . 4 |- `'(A o. B) = (`'B o. `'A)
12	4, 10, 11	3eqtr4g 1531	. . 3 |- (dom A (_ C -> `'(A o. `'(`'B |` C)) = `'(A o. B))
13		cnveq 3292	. . 3 |- (`'(A o. `'(`'B |` C)) = `'(A o. B) -> `'`'(A o. `'(`'B |` C)) = `'`'(A o. B))
14	12, 13	syl 10	. 2 |- (dom A (_ C -> `'`'(A o. `'(`'B |` C)) = `'`'(A o. B))
15		relco 3484	. . 3 |- Rel (A o. `'(`'B |` C))
16		dfrel2 3485	. . 3 |- (Rel (A o. `'(`'B |` C)) <-> `'`'(A o. `'(`'B |` C)) = (A o. `'(`'B |` C)))
17	15, 16	mpbi 189	. 2 |- `'`'(A o. `'(`'B |` C)) = (A o. `'(`'B |` C))
18		relco 3484	. . 3 |- Rel (A o. B)
19		dfrel2 3485	. . 3 |- (Rel (A o. B) <-> `'`'(A o. B) = (A o. B))
20	18, 19	mpbi 189	. 2 |- `'`'(A o. B) = (A o. B)
21	14, 17, 20	3eqtr3g 1530	1 |- (dom A (_ C -> (A o. `'(`'B |` C)) = (A o. B))

Syntax hints:   -> wi 3   = wceq 956   (_ wss 2047  `'ccnv 3169  dom cdm 3170  ran crn 3171   |` cres 3172   o. ccom 3174  Rel wrel 3175

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190

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