Theorem nelrnres 39374

Description: If A is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
nelrnres	|- ( -. A e. ran B -> -. A e. ran ( B |` C ) )

Proof of Theorem nelrnres

Step	Hyp	Ref	Expression
1		rnresss 39365	. 2 |- ran ( B |` C ) C_ ran B
2		ssnel 39204	. 2 |- ( ( ran ( B |` C ) C_ ran B /\ -. A e. ran B ) -> -. A e. ran ( B |` C ) )
3	1, 2	mpan 706	1 |- ( -. A e. ran B -> -. A e. ran ( B |` C ) )

Syntax hints:   -. wn 3   -> wi 4   e. wcel 1990   C_ wss 3574   ran crn 5115   |` 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

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-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126

This theorem is referenced by:  sge0sup  40608  sge0less  40609  sge0resplit  40623