Theorem resmpt 5086

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 5085	. 2 |- ( B C_ A -> ( { <. x , y >. | ( x e. A /\ y = C ) } |` B ) = { <. x , y >. | ( x e. B /\ y = C ) } )
2		df-mpt 4173	. . 3 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
3	2	reseq1i 5034	. 2 |- ( ( x e. A |-> C ) |` B ) = ( { <. x , y >. | ( x e. A /\ y = C ) } |` B )
4		df-mpt 4173	. 2 |- ( x e. B |-> C ) = { <. x , y >. | ( x e. B /\ y = C ) }
5	1, 3, 4	3eqtr4g 2290	1 |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   C_ wss 3211   {copab 4170   |-> cmpt 4171   |` cres 4751

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-opab 4172  df-mpt 4173  df-xp 4755  df-rel 4756  df-res 4761

This theorem is referenced by:  resmpt3  5087  resmptf  5088  resmptd  5089  f1stres  6353  f2ndres  6354  tposss  6477  dftpos2  6492  dftpos4  6494  djuf1olemr  7345  fisumss  12078  isumclim3  12109  expcnv  12190  fprodssdc  12276  conjsubg  13994  gsumfzfsumlemm  14735  tgrest  15034  cnmptid  15146  hovercncf  15511  dvidlemap  15556  dvidrelem  15557  dvidsslem  15558  dvcnp2cntop  15564  dvmulxxbr  15567  dvcoapbr  15572  dvrecap  15578