MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsup Structured version   Unicode version

Theorem nfsup 7923
Description: Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
Hypotheses
Ref Expression
nfsup.1  |-  F/_ x A
nfsup.2  |-  F/_ x B
nfsup.3  |-  F/_ x R
Assertion
Ref Expression
nfsup  |-  F/_ x sup ( A ,  B ,  R )

Proof of Theorem nfsup
StepHypRef Expression
1 dfsup2 7914 . 2  |-  sup ( A ,  B ,  R )  =  U. ( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
2 nfsup.2 . . . 4  |-  F/_ x B
3 nfsup.3 . . . . . . 7  |-  F/_ x R
43nfcnv 5187 . . . . . 6  |-  F/_ x `' R
5 nfsup.1 . . . . . 6  |-  F/_ x A
64, 5nfima 5351 . . . . 5  |-  F/_ x
( `' R " A )
72, 6nfdif 3630 . . . . . 6  |-  F/_ x
( B  \  ( `' R " A ) )
83, 7nfima 5351 . . . . 5  |-  F/_ x
( R " ( B  \  ( `' R " A ) ) )
96, 8nfun 3665 . . . 4  |-  F/_ x
( ( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) )
102, 9nfdif 3630 . . 3  |-  F/_ x
( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
1110nfuni 4257 . 2  |-  F/_ x U. ( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
121, 11nfcxfr 2627 1  |-  F/_ x sup ( A ,  B ,  R )
Colors of variables: wff setvar class
Syntax hints:   F/_wnfc 2615    \ cdif 3478    u. cun 3479   U.cuni 4251   `'ccnv 5004   "cima 5008   supcsup 7912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-sup 7913
This theorem is referenced by:  iundisj  21826  itg2cnlem1  22036  iundisjf  27262  iundisjfi  27415  nfwsuc  29292  nfwlim  29296  totbndbnd  30203  aomclem8  30926  ssfiunibd  31400  stoweidlem62  31676  fourierdlem20  31741  fourierdlem31  31752  fourierdlem79  31800
  Copyright terms: Public domain W3C validator