Theorem resmptd 4997

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 4994	. 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 1364   C_ wss 3157   |-> cmpt 4094   |` cres 4665

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-mpt 4096  df-xp 4669  df-rel 4670  df-res 4675

This theorem is referenced by:  fisumss  11557  fprodssdc  11755  conjsubgen  13408  gsumfzconst  13471  cnmpt1res  14532  bj-charfundcALT  15455