Theorem resmpt 5061

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 5060	. 2 |- ( B C_ A -> ( { <. x , y >. | ( x e. A /\ y = C ) } |` B ) = { <. x , y >. | ( x e. B /\ y = C ) } )
2		df-mpt 4152	. . 3 |- ( x e. A |-> C ) = { <. x , y >. | ( x e. A /\ y = C ) }
3	2	reseq1i 5009	. 2 |- ( ( x e. A |-> C ) |` B ) = ( { <. x , y >. | ( x e. A /\ y = C ) } |` B )
4		df-mpt 4152	. 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 1397   e. wcel 2202   C_ wss 3200   {copab 4149   |-> cmpt 4150   |` cres 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-res 4737

This theorem is referenced by:  resmpt3  5062  resmptf  5063  resmptd  5064  f1stres  6322  f2ndres  6323  tposss  6412  dftpos2  6427  dftpos4  6429  djuf1olemr  7253  fisumss  11971  isumclim3  12002  expcnv  12083  fprodssdc  12169  conjsubg  13882  gsumfzfsumlemm  14620  tgrest  14912  cnmptid  15024  hovercncf  15389  dvidlemap  15434  dvidrelem  15435  dvidsslem  15436  dvcnp2cntop  15442  dvmulxxbr  15445  dvcoapbr  15450  dvrecap  15456