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Theorem nfsup 8903
Description: Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.)
Hypotheses
Ref Expression
nfsup.1 𝑥𝐴
nfsup.2 𝑥𝐵
nfsup.3 𝑥𝑅
Assertion
Ref Expression
nfsup 𝑥sup(𝐴, 𝐵, 𝑅)

Proof of Theorem nfsup
StepHypRef Expression
1 dfsup2 8896 . 2 sup(𝐴, 𝐵, 𝑅) = (𝐵 ∖ ((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴)))))
2 nfsup.2 . . . 4 𝑥𝐵
3 nfsup.3 . . . . . . 7 𝑥𝑅
43nfcnv 5742 . . . . . 6 𝑥𝑅
5 nfsup.1 . . . . . 6 𝑥𝐴
64, 5nfima 5930 . . . . 5 𝑥(𝑅𝐴)
72, 6nfdif 4099 . . . . . 6 𝑥(𝐵 ∖ (𝑅𝐴))
83, 7nfima 5930 . . . . 5 𝑥(𝑅 “ (𝐵 ∖ (𝑅𝐴)))
96, 8nfun 4138 . . . 4 𝑥((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴))))
102, 9nfdif 4099 . . 3 𝑥(𝐵 ∖ ((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴)))))
1110nfuni 4837 . 2 𝑥 (𝐵 ∖ ((𝑅𝐴) ∪ (𝑅 “ (𝐵 ∖ (𝑅𝐴)))))
121, 11nfcxfr 2972 1 𝑥sup(𝐴, 𝐵, 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wnfc 2958  cdif 3930  cun 3931   cuni 4830  ccnv 5547  cima 5551  supcsup 8892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-sup 8894
This theorem is referenced by:  nfinf  8934  itg2cnlem1  24289  esum2d  31251  nfwlim  33006  totbndbnd  34948  aomclem8  39539  binomcxplemdvbinom  40562  binomcxplemdvsum  40564  binomcxplemnotnn0  40565  ssfiunibd  41452  uzub  41581  limsupubuz  41870  fourierdlem20  42289  fourierdlem31  42300  fourierdlem79  42347  sge0ltfirp  42559  pimdecfgtioc  42870  decsmflem  42919  smfsup  42965  smfsupxr  42967  smflimsup  42979
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