Theorem resmpt 5067

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 5066	. 2 |- ( B C_ A -> ( { <. x , y >. | ( x e. A /\ y = C ) } |` B ) = { <. x , y >. | ( x e. B /\ y = C ) } )
2		df-mpt 4157	. . 3 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
3	2	reseq1i 5015	. 2 |- ( ( x e. A |-> C ) |` B ) = ( { <. x , y >. | ( x e. A /\ y = C ) } |` B )
4		df-mpt 4157	. 2 |- ( x e. B |-> C ) = { <. x , y >. | ( x e. B /\ y = C ) }
5	1, 3, 4	3eqtr4g 2289	1 |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   C_ wss 3201   {copab 4154   |-> cmpt 4155   |` cres 4733

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-mpt 4157  df-xp 4737  df-rel 4738  df-res 4743

This theorem is referenced by:  resmpt3  5068  resmptf  5069  resmptd  5070  f1stres  6331  f2ndres  6332  tposss  6455  dftpos2  6470  dftpos4  6472  djuf1olemr  7296  fisumss  12016  isumclim3  12047  expcnv  12128  fprodssdc  12214  conjsubg  13927  gsumfzfsumlemm  14666  tgrest  14963  cnmptid  15075  hovercncf  15440  dvidlemap  15485  dvidrelem  15486  dvidsslem  15487  dvcnp2cntop  15493  dvmulxxbr  15496  dvcoapbr  15501  dvrecap  15507