Theorem resmptd 5070

Description: Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypothesis

Ref	Expression
resmptd.b	|- ( ph -> B C_ A )

Assertion

Ref	Expression
resmptd	|- ( ph -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   C( x)

Proof of Theorem resmptd

Step	Hyp	Ref	Expression
1		resmptd.b	. 2 |- ( ph -> B C_ A )
2		resmpt 5067	. 2 |- ( B C_ A -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )
3	1, 2	syl 14	1 |- ( ph -> ( ( x e. A |-> C ) |` B ) = ( x e. B |-> C ) )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3201   |-> 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:  fisumss  12016  fprodssdc  12214  conjsubgen  13928  gsumfzconst  13991  cnmpt1res  15090  bj-charfundcALT  16508