Theorem wunres 9553

Description: A weak universe is closed under restrictions. (Contributed by Mario Carneiro, 12-Jan-2017.)

Hypotheses

Ref	Expression
wun0.1	|- ( ph -> U e. WUni )
wunop.2	|- ( ph -> A e. U )

Assertion

Ref	Expression
wunres	|- ( ph -> ( A |` B ) e. U )

Proof of Theorem wunres

Step	Hyp	Ref	Expression
1		wun0.1	. 2 |- ( ph -> U e. WUni )
2		wunop.2	. 2 |- ( ph -> A e. U )
3		resss 5422	. . 3 |- ( A |` B ) C_ A
4	3	a1i 11	. 2 |- ( ph -> ( A |` B ) C_ A )
5	1, 2, 4	wunss 9534	1 |- ( ph -> ( A |` B ) e. U )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   |` cres 5116  WUnicwun 9522

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

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-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-in 3581  df-ss 3588  df-pw 4160  df-uni 4437  df-tr 4753  df-res 5126  df-wun 9524

This theorem is referenced by:  wunsets  15900