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

Theorem nfsup 7292
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 7285 . 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 4942 . . . . . 6  |-  F/_ x `' R
5 nfsup.1 . . . . . 6  |-  F/_ x A
64, 5nfima 5102 . . . . 5  |-  F/_ x
( `' R " A )
72, 6nfdif 3373 . . . . . 6  |-  F/_ x
( B  \  ( `' R " A ) )
83, 7nfima 5102 . . . . 5  |-  F/_ x
( R " ( B  \  ( `' R " A ) ) )
96, 8nfun 3407 . . . 4  |-  F/_ x
( ( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) )
102, 9nfdif 3373 . . 3  |-  F/_ x
( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
1110nfuni 3914 . 2  |-  F/_ x U. ( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
121, 11nfcxfr 2491 1  |-  F/_ x sup ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2481    \ cdif 3225    u. cun 3226   U.cuni 3908   `'ccnv 4770   "cima 4774   supcsup 7283
This theorem is referenced by:  iundisj  19009  itg2cnlem1  19220  iundisjf  23227  iundisjfi  23356  totbndbnd  25836  aomclem8  26482  stoweidlem62  27134
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-xp 4777  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-sup 7284
  Copyright terms: Public domain W3C validator