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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 9376 | . 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 5832 | . . . . . 6 ⊢ Ⅎ𝑥◡𝑅 |
5 | nfsup.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfima 6019 | . . . . 5 ⊢ Ⅎ𝑥(◡𝑅 “ 𝐴) |
7 | 2, 6 | nfdif 4083 | . . . . . 6 ⊢ Ⅎ𝑥(𝐵 ∖ (◡𝑅 “ 𝐴)) |
8 | 3, 7 | nfima 6019 | . . . . 5 ⊢ Ⅎ𝑥(𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))) |
9 | 6, 8 | nfun 4123 | . . . 4 ⊢ Ⅎ𝑥((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴)))) |
10 | 2, 9 | nfdif 4083 | . . 3 ⊢ Ⅎ𝑥(𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) |
11 | 10 | nfuni 4870 | . 2 ⊢ Ⅎ𝑥∪ (𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) |
12 | 1, 11 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2885 ∖ cdif 3905 ∪ cun 3906 ∪ cuni 4863 ◡ccnv 5630 “ cima 5634 supcsup 9372 |
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 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
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 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-xp 5637 df-cnv 5639 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-sup 9374 |
This theorem is referenced by: nfinf 9414 itg2cnlem1 25110 esum2d 32561 nfwlim 34267 totbndbnd 36215 aomclem8 41326 binomcxplemdvbinom 42575 binomcxplemdvsum 42577 binomcxplemnotnn0 42578 ssfiunibd 43479 uzub 43602 limsupubuz 43886 fourierdlem20 44300 fourierdlem31 44311 fourierdlem79 44358 sge0ltfirp 44573 pimdecfgtioc 44888 decsmflem 44939 smfsup 44987 smfsupxr 44989 smflimsup 45001 |
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