Theorem cores 4824

Description: Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
cores	|- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) )

Proof of Theorem cores

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2560	. . . . . . 7 |- z e. _V
2		vex 2560	. . . . . . 7 |- y e. _V
3	1, 2	brelrn 4567	. . . . . 6 |- ( z B y -> y e. ran B )
4		ssel 2939	. . . . . 6 |- ( ran B C_ C -> ( y e. ran B -> y e. C ) )
5		vex 2560	. . . . . . . 8 |- x e. _V
6	5	brres 4618	. . . . . . 7 |- ( y ( A |` C ) x <-> ( y A x /\ y e. C ) )
7	6	rbaib 830	. . . . . 6 |- ( y e. C -> ( y ( A |` C ) x <-> y A x ) )
8	3, 4, 7	syl56 30	. . . . 5 |- ( ran B C_ C -> ( z B y -> ( y ( A |` C ) x <-> y A x ) ) )
9	8	pm5.32d 423	. . . 4 |- ( ran B C_ C -> ( ( z B y /\ y ( A |` C ) x ) <-> ( z B y /\ y A x ) ) )
10	9	exbidv 1706	. . 3 |- ( ran B C_ C -> ( E. y ( z B y /\ y ( A |` C ) x ) <-> E. y ( z B y /\ y A x ) ) )
11	10	opabbidv 3823	. 2 |- ( ran B C_ C -> { <. z , x >. | E. y ( z B y /\ y ( A |` C ) x ) } = { <. z , x >. | E. y ( z B y /\ y A x ) } )
12		df-co 4354	. 2 |- ( ( A |` C ) o. B ) = { <. z , x >. | E. y ( z B y /\ y ( A |` C ) x ) }
13		df-co 4354	. 2 |- ( A o. B ) = { <. z , x >. | E. y ( z B y /\ y A x ) }
14	11, 12, 13	3eqtr4g 2097	1 |- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   e. wcel 1393   C_ wss 2917   class class class wbr 3764   {copab 3817   ran crn 4346   |` cres 4347   o. ccom 4349

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357

This theorem is referenced by:  cocnvcnv1  4831  cores2  4833  relcoi2  4848  fco2  5057  fcoi2  5071