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Mirrors > Home > MPE Home > Th. List > nfsup | Structured version Visualization version GIF version |
Description: Hypothesis builder for supremum. (Contributed by Mario Carneiro, 20-Mar-2014.) |
Ref | Expression |
---|---|
nfsup.1 | ⊢ Ⅎ𝑥𝐴 |
nfsup.2 | ⊢ Ⅎ𝑥𝐵 |
nfsup.3 | ⊢ Ⅎ𝑥𝑅 |
Ref | Expression |
---|---|
nfsup | ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfsup2 8892 | . 2 ⊢ sup(𝐴, 𝐵, 𝑅) = ∪ (𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) | |
2 | nfsup.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
3 | nfsup.3 | . . . . . . 7 ⊢ Ⅎ𝑥𝑅 | |
4 | 3 | nfcnv 5713 | . . . . . 6 ⊢ Ⅎ𝑥◡𝑅 |
5 | nfsup.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfima 5904 | . . . . 5 ⊢ Ⅎ𝑥(◡𝑅 “ 𝐴) |
7 | 2, 6 | nfdif 4053 | . . . . . 6 ⊢ Ⅎ𝑥(𝐵 ∖ (◡𝑅 “ 𝐴)) |
8 | 3, 7 | nfima 5904 | . . . . 5 ⊢ Ⅎ𝑥(𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))) |
9 | 6, 8 | nfun 4092 | . . . 4 ⊢ Ⅎ𝑥((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴)))) |
10 | 2, 9 | nfdif 4053 | . . 3 ⊢ Ⅎ𝑥(𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) |
11 | 10 | nfuni 4807 | . 2 ⊢ Ⅎ𝑥∪ (𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) |
12 | 1, 11 | nfcxfr 2953 | 1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2936 ∖ cdif 3878 ∪ cun 3879 ∪ cuni 4800 ◡ccnv 5518 “ cima 5522 supcsup 8888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-sup 8890 |
This theorem is referenced by: nfinf 8930 itg2cnlem1 24365 esum2d 31462 nfwlim 33222 totbndbnd 35227 aomclem8 40005 binomcxplemdvbinom 41057 binomcxplemdvsum 41059 binomcxplemnotnn0 41060 ssfiunibd 41941 uzub 42068 limsupubuz 42355 fourierdlem20 42769 fourierdlem31 42780 fourierdlem79 42827 sge0ltfirp 43039 pimdecfgtioc 43350 decsmflem 43399 smfsup 43445 smfsupxr 43447 smflimsup 43459 |
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