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Theorem nfsup 7416
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 7409 . 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 5014 . . . . . 6  |-  F/_ x `' R
5 nfsup.1 . . . . . 6  |-  F/_ x A
64, 5nfima 5174 . . . . 5  |-  F/_ x
( `' R " A )
72, 6nfdif 3432 . . . . . 6  |-  F/_ x
( B  \  ( `' R " A ) )
83, 7nfima 5174 . . . . 5  |-  F/_ x
( R " ( B  \  ( `' R " A ) ) )
96, 8nfun 3467 . . . 4  |-  F/_ x
( ( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) )
102, 9nfdif 3432 . . 3  |-  F/_ x
( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
1110nfuni 3985 . 2  |-  F/_ x U. ( B  \  (
( `' R " A )  u.  ( R " ( B  \ 
( `' R " A ) ) ) ) )
121, 11nfcxfr 2541 1  |-  F/_ x sup ( A ,  B ,  R )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2531    \ cdif 3281    u. cun 3282   U.cuni 3979   `'ccnv 4840   "cima 4844   supcsup 7407
This theorem is referenced by:  iundisj  19399  itg2cnlem1  19610  iundisjf  23986  iundisjfi  24109  totbndbnd  26392  aomclem8  27031  stoweidlem62  27682
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-xp 4847  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-sup 7408
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