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Theorem nfsup 7456
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 7447 . 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 5051 . . . . . 6  |-  F/_ x `' R
5 nfsup.1 . . . . . 6  |-  F/_ x A
64, 5nfima 5211 . . . . 5  |-  F/_ x
( `' R " A )
72, 6nfdif 3468 . . . . . 6  |-  F/_ x
( B  \  ( `' R " A ) )
83, 7nfima 5211 . . . . 5  |-  F/_ x
( R " ( B  \  ( `' R " A ) ) )
96, 8nfun 3503 . . . 4  |-  F/_ x
( ( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) )
102, 9nfdif 3468 . . 3  |-  F/_ x
( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
1110nfuni 4021 . 2  |-  F/_ x U. ( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
121, 11nfcxfr 2569 1  |-  F/_ x sup ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2559    \ cdif 3317    u. cun 3318   U.cuni 4015   `'ccnv 4877   "cima 4881   supcsup 7445
This theorem is referenced by:  iundisj  19442  itg2cnlem1  19653  iundisjf  24029  iundisjfi  24152  nfwsuc  25569  nfwlim  25573  totbndbnd  26498  aomclem8  27136  stoweidlem62  27787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-sup 7446
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