Theorem cores 5286

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 2824	. . . . . . 7 |- z e. _V
2		vex 2824	. . . . . . 7 |- y e. _V
3	1, 2	brelrn 5010	. . . . . 6 |- ( z B y -> y e. ran B )
4		ssel 3242	. . . . . 6 |- ( ran B C_ C -> ( y e. ran B -> y e. C ) )
5		vex 2824	. . . . . . . 8 |- x e. _V
6	5	brres 5064	. . . . . . 7 |- ( y ( A |` C ) x <-> ( y A x /\ y e. C ) )
7	6	rbaib 933	. . . . . 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 454	. . . 4 |- ( ran B C_ C -> ( ( z B y /\ y ( A |` C ) x ) <-> ( z B y /\ y A x ) ) )
10	9	exbidv 1878	. . 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 4192	. 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 4778	. 2 |- ( ( A |` C ) o. B ) = { <. z , x >. | E. y ( z B y /\ y ( A |` C ) x ) }
13		df-co 4778	. 2 |- ( A o. B ) = { <. z , x >. | E. y ( z B y /\ y A x ) }
14	11, 12, 13	3eqtr4g 2296	1 |- ( ran B C_ C -> ( ( A |` C ) o. B ) = ( A o. B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   E.wex 1545   e. wcel 2209   C_ wss 3220   class class class wbr 4125   {copab 4186   ran crn 4770   |` cres 4771   o. ccom 4773

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781

This theorem is referenced by:  cocnvcnv1  5293  cores2  5295  cocnvres  5307  relcoi2  5313  fco2  5549  fcoi2  5568  gsump1  14134