Theorem cores 5124

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 2738	. . . . . . 7 |- z e. _V
2		vex 2738	. . . . . . 7 |- y e. _V
3	1, 2	brelrn 4853	. . . . . 6 |- ( z B y -> y e. ran B )
4		ssel 3147	. . . . . 6 |- ( ran B C_ C -> ( y e. ran B -> y e. C ) )
5		vex 2738	. . . . . . . 8 |- x e. _V
6	5	brres 4906	. . . . . . 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 1823	. . 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 4064	. 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 4629	. 2 |- ( ( A |` C ) o. B ) = { <. z , x >. | E. y ( z B y /\ y ( A |` C ) x ) }
13		df-co 4629	. 2 |- ( A o. B ) = { <. z , x >. | E. y ( z B y /\ y A x ) }
14	11, 12, 13	3eqtr4g 2233	1 |- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   E.wex 1490   e. wcel 2146   C_ wss 3127   class class class wbr 3998   {copab 4058   ran crn 4621   |` cres 4622   o. ccom 4624

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632

This theorem is referenced by:  cocnvcnv1  5131  cores2  5133  cocnvres  5145  relcoi2  5151  fco2  5374  fcoi2  5389