Theorem resmpo 6129

Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)

Assertion

Ref	Expression
resmpo	|- ( ( C C_ A /\ D C_ B ) -> ( ( x e. A , y e. B |-> E ) |` ( C X. D ) ) = ( x e. C , y e. D |-> E ) )

Distinct variable groups:   x, A, y   x, B, y   x, C, y   x, D, y

Allowed substitution hints:   E( x, y)

Proof of Theorem resmpo

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resoprab2 6128	. 2 |- ( ( C C_ A /\ D C_ B ) -> ( { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = E ) } |` ( C X. D ) ) = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ z = E ) } )
2		df-mpo 6033	. . 3 |- ( x e. A , y e. B |-> E ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = E ) }
3	2	reseq1i 5015	. 2 |- ( ( x e. A , y e. B |-> E ) |` ( C X. D ) ) = ( { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = E ) } |` ( C X. D ) )
4		df-mpo 6033	. 2 |- ( x e. C , y e. D |-> E ) = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ z = E ) }
5	1, 3, 4	3eqtr4g 2289	1 |- ( ( C C_ A /\ D C_ B ) -> ( ( x e. A , y e. B |-> E ) |` ( C X. D ) ) = ( x e. C , y e. D |-> E ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   C_ wss 3201   X. cxp 4729   |` cres 4733   {coprab 6029   e. cmpo 6030

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-xp 4737  df-rel 4738  df-res 4743  df-oprab 6032  df-mpo 6033

This theorem is referenced by:  ofmres  6307  divfnzn  9916  txss12  15077  txbasval  15078  cnmpt2res  15108