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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 9485 | . 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 5888 | . . . . . 6 ⊢ Ⅎ𝑥◡𝑅 | 
| 5 | nfsup.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
| 6 | 4, 5 | nfima 6085 | . . . . 5 ⊢ Ⅎ𝑥(◡𝑅 “ 𝐴) | 
| 7 | 2, 6 | nfdif 4128 | . . . . . 6 ⊢ Ⅎ𝑥(𝐵 ∖ (◡𝑅 “ 𝐴)) | 
| 8 | 3, 7 | nfima 6085 | . . . . 5 ⊢ Ⅎ𝑥(𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))) | 
| 9 | 6, 8 | nfun 4169 | . . . 4 ⊢ Ⅎ𝑥((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴)))) | 
| 10 | 2, 9 | nfdif 4128 | . . 3 ⊢ Ⅎ𝑥(𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) | 
| 11 | 10 | nfuni 4913 | . 2 ⊢ Ⅎ𝑥∪ (𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) | 
| 12 | 1, 11 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) | 
| Colors of variables: wff setvar class | 
| Syntax hints: Ⅎwnfc 2889 ∖ cdif 3947 ∪ cun 3948 ∪ cuni 4906 ◡ccnv 5683 “ cima 5687 supcsup 9481 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-sup 9483 | 
| This theorem is referenced by: nfinf 9523 itg2cnlem1 25797 esum2d 34095 nfwlim 35824 totbndbnd 37797 aomclem8 43078 binomcxplemdvbinom 44377 binomcxplemdvsum 44379 binomcxplemnotnn0 44380 ssfiunibd 45326 uzub 45447 limsupubuz 45733 fourierdlem20 46147 fourierdlem31 46158 fourierdlem79 46205 sge0ltfirp 46420 pimdecfgtioc 46735 decsmflem 46786 smfsup 46834 smfsupxr 46836 smflimsup 46848 | 
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