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Theorem nfsup 7807
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 7798 . 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 5121 . . . . . 6  |-  F/_ x `' R
5 nfsup.1 . . . . . 6  |-  F/_ x A
64, 5nfima 5280 . . . . 5  |-  F/_ x
( `' R " A )
72, 6nfdif 3580 . . . . . 6  |-  F/_ x
( B  \  ( `' R " A ) )
83, 7nfima 5280 . . . . 5  |-  F/_ x
( R " ( B  \  ( `' R " A ) ) )
96, 8nfun 3615 . . . 4  |-  F/_ x
( ( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) )
102, 9nfdif 3580 . . 3  |-  F/_ x
( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
1110nfuni 4200 . 2  |-  F/_ x U. ( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
121, 11nfcxfr 2612 1  |-  F/_ x sup ( A ,  B ,  R )
Colors of variables: wff setvar class
Syntax hints:   F/_wnfc 2600    \ cdif 3428    u. cun 3429   U.cuni 4194   `'ccnv 4942   "cima 4946   supcsup 7796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-xp 4949  df-cnv 4951  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-sup 7797
This theorem is referenced by:  iundisj  21157  itg2cnlem1  21367  iundisjf  26077  iundisjfi  26220  nfwsuc  27894  nfwlim  27898  totbndbnd  28831  aomclem8  29557  stoweidlem62  30000
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