![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9438 | . 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 5878 | . . . . . 6 ⊢ Ⅎ𝑥◡𝑅 |
5 | nfsup.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfima 6067 | . . . . 5 ⊢ Ⅎ𝑥(◡𝑅 “ 𝐴) |
7 | 2, 6 | nfdif 4125 | . . . . . 6 ⊢ Ⅎ𝑥(𝐵 ∖ (◡𝑅 “ 𝐴)) |
8 | 3, 7 | nfima 6067 | . . . . 5 ⊢ Ⅎ𝑥(𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))) |
9 | 6, 8 | nfun 4165 | . . . 4 ⊢ Ⅎ𝑥((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴)))) |
10 | 2, 9 | nfdif 4125 | . . 3 ⊢ Ⅎ𝑥(𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) |
11 | 10 | nfuni 4915 | . 2 ⊢ Ⅎ𝑥∪ (𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) |
12 | 1, 11 | nfcxfr 2901 | 1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2883 ∖ cdif 3945 ∪ cun 3946 ∪ cuni 4908 ◡ccnv 5675 “ cima 5679 supcsup 9434 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-xp 5682 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-sup 9436 |
This theorem is referenced by: nfinf 9476 itg2cnlem1 25278 esum2d 33086 nfwlim 34789 totbndbnd 36652 aomclem8 41793 binomcxplemdvbinom 43102 binomcxplemdvsum 43104 binomcxplemnotnn0 43105 ssfiunibd 44009 uzub 44131 limsupubuz 44419 fourierdlem20 44833 fourierdlem31 44844 fourierdlem79 44891 sge0ltfirp 45106 pimdecfgtioc 45421 decsmflem 45472 smfsup 45520 smfsupxr 45522 smflimsup 45534 |
Copyright terms: Public domain | W3C validator |