Theorem resmptd 4994

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 4991	. 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 3154   |-> cmpt 4091   |` cres 4662

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-opab 4092  df-mpt 4093  df-xp 4666  df-rel 4667  df-res 4672

This theorem is referenced by:  fisumss  11538  fprodssdc  11736  conjsubgen  13351  gsumfzconst  13414  cnmpt1res  14475  bj-charfundcALT  15371