Theorem mptss 5454

Description: Sufficient condition for inclusion in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
mptss	|- ( A C_ B -> ( x e. A |-> C ) C_ ( x e. B |-> C ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem mptss

Step	Hyp	Ref	Expression
1		resmpt 5449	. 2 |- ( A C_ B -> ( ( x e. B |-> C ) |` A ) = ( x e. A |-> C ) )
2		resss 5422	. 2 |- ( ( x e. B |-> C ) |` A ) C_ ( x e. B |-> C )
3	1, 2	syl6eqssr 3656	1 |- ( A C_ B -> ( x e. A |-> C ) C_ ( x e. B |-> C ) )

Syntax hints:   -> wi 4   C_ wss 3574   |-> cmpt 4729   |` cres 5116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-mpt 4730  df-xp 5120  df-rel 5121  df-res 5126

This theorem is referenced by:  carsgclctunlem2  30381  sge0less  40609