Theorem cores 5134

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 2742	. . . . . . 7 |- z e. _V
2		vex 2742	. . . . . . 7 |- y e. _V
3	1, 2	brelrn 4862	. . . . . 6 |- ( z B y -> y e. ran B )
4		ssel 3151	. . . . . 6 |- ( ran B C_ C -> ( y e. ran B -> y e. C ) )
5		vex 2742	. . . . . . . 8 |- x e. _V
6	5	brres 4915	. . . . . . 7 |- ( y ( A |` C ) x <-> ( y A x /\ y e. C ) )
7	6	rbaib 921	. . . . . 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 450	. . . 4 |- ( ran B C_ C -> ( ( z B y /\ y ( A |` C ) x ) <-> ( z B y /\ y A x ) ) )
10	9	exbidv 1825	. . 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 4071	. 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 4637	. 2 |- ( ( A |` C ) o. B ) = { <. z , x >. | E. y ( z B y /\ y ( A |` C ) x ) }
13		df-co 4637	. 2 |- ( A o. B ) = { <. z , x >. | E. y ( z B y /\ y A x ) }
14	11, 12, 13	3eqtr4g 2235	1 |- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1492   e. wcel 2148   C_ wss 3131   class class class wbr 4005   {copab 4065   ran crn 4629   |` cres 4630   o. ccom 4632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640

This theorem is referenced by:  cocnvcnv1  5141  cores2  5143  cocnvres  5155  relcoi2  5161  fco2  5384  fcoi2  5399