Theorem resmpt 4760

Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)

Assertion

Ref	Expression
resmpt	|- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem resmpt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resopab2 4759	. 2 |- ( B C_ A -> ( { <. x , y >. | ( x e. A /\ y = C ) } |` B ) = { <. x , y >. | ( x e. B /\ y = C ) } )
2		df-mpt 3901	. . 3 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
3	2	reseq1i 4709	. 2 |- ( ( x e. A |-> C ) |` B ) = ( { <. x , y >. | ( x e. A /\ y = C ) } |` B )
4		df-mpt 3901	. 2 |- ( x e. B |-> C ) = { <. x , y >. | ( x e. B /\ y = C ) }
5	1, 3, 4	3eqtr4g 2145	1 |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   C_ wss 2999   {copab 3898   |-> cmpt 3899   |` cres 4440

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-mpt 3901  df-xp 4444  df-rel 4445  df-res 4450

This theorem is referenced by:  resmpt3  4761  resmptf  4762  resmptd  4763  f1stres  5930  f2ndres  5931  tposss  6011  dftpos2  6026  dftpos4  6028  djuf1olemr  6746  fisumss  10784  isumclim3  10817  expcnv  10898