Theorem cores 4844

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 2604	. . . . . . 7 |- z e. _V
2		vex 2604	. . . . . . 7 |- y e. _V
3	1, 2	brelrn 4585	. . . . . 6 |- ( z B y -> y e. ran B )
4		ssel 2993	. . . . . 6 |- ( ran B C_ C -> ( y e. ran B -> y e. C ) )
5		vex 2604	. . . . . . . 8 |- x e. _V
6	5	brres 4636	. . . . . . 7 |- ( y ( A |` C ) x <-> ( y A x /\ y e. C ) )
7	6	rbaib 863	. . . . . 6 |- ( y e. C -> ( y ( A |` C ) x <-> y A x ) )
8	3, 4, 7	syl56 34	. . . . 5 |- ( ran B C_ C -> ( z B y -> ( y ( A |` C ) x <-> y A x ) ) )
9	8	pm5.32d 437	. . . 4 |- ( ran B C_ C -> ( ( z B y /\ y ( A |` C ) x ) <-> ( z B y /\ y A x ) ) )
10	9	exbidv 1746	. . 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 3844	. 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 4372	. 2 |- ( ( A |` C ) o. B ) = { <. z , x >. | E. y ( z B y /\ y ( A |` C ) x ) }
13		df-co 4372	. 2 |- ( A o. B ) = { <. z , x >. | E. y ( z B y /\ y A x ) }
14	11, 12, 13	3eqtr4g 2138	1 |- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   C_ wss 2973   class class class wbr 3785   {copab 3838   ran crn 4364   |` cres 4365   o. ccom 4367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375

This theorem is referenced by:  cocnvcnv1  4851  cores2  4853  relcoi2  4868  fco2  5077  fcoi2  5091