Theorem resmpo 6114

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 6113	. 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 6018	. . 3 |- ( x e. A , y e. B |-> E ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = E ) }
3	2	reseq1i 5007	. 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 6018	. 2 |- ( x e. C , y e. D |-> E ) = { <. <. x , y >. , z >. | ( ( x e. C /\ y e. D ) /\ z = E ) }
5	1, 3, 4	3eqtr4g 2287	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 1395   e. wcel 2200   C_ wss 3198   X. cxp 4721   |` cres 4725   {coprab 6014   e. cmpo 6015

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-opab 4149  df-xp 4729  df-rel 4730  df-res 4735  df-oprab 6017  df-mpo 6018

This theorem is referenced by:  ofmres  6293  divfnzn  9845  txss12  14980  txbasval  14981  cnmpt2res  15011