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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 9482 | . 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 5892 | . . . . . 6 ⊢ Ⅎ𝑥◡𝑅 |
5 | nfsup.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfima 6088 | . . . . 5 ⊢ Ⅎ𝑥(◡𝑅 “ 𝐴) |
7 | 2, 6 | nfdif 4139 | . . . . . 6 ⊢ Ⅎ𝑥(𝐵 ∖ (◡𝑅 “ 𝐴)) |
8 | 3, 7 | nfima 6088 | . . . . 5 ⊢ Ⅎ𝑥(𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))) |
9 | 6, 8 | nfun 4180 | . . . 4 ⊢ Ⅎ𝑥((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴)))) |
10 | 2, 9 | nfdif 4139 | . . 3 ⊢ Ⅎ𝑥(𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) |
11 | 10 | nfuni 4919 | . 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 2888 ∖ cdif 3960 ∪ cun 3961 ∪ cuni 4912 ◡ccnv 5688 “ cima 5692 supcsup 9478 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-cnv 5697 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-sup 9480 |
This theorem is referenced by: nfinf 9520 itg2cnlem1 25811 esum2d 34074 nfwlim 35804 totbndbnd 37776 aomclem8 43050 binomcxplemdvbinom 44349 binomcxplemdvsum 44351 binomcxplemnotnn0 44352 ssfiunibd 45260 uzub 45381 limsupubuz 45669 fourierdlem20 46083 fourierdlem31 46094 fourierdlem79 46141 sge0ltfirp 46356 pimdecfgtioc 46671 decsmflem 46722 smfsup 46770 smfsupxr 46772 smflimsup 46784 |
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