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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 8233 | . 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 5223 | . . . . . 6 ⊢ Ⅎ𝑥◡𝑅 |
5 | nfsup.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfima 5393 | . . . . 5 ⊢ Ⅎ𝑥(◡𝑅 “ 𝐴) |
7 | 2, 6 | nfdif 3693 | . . . . . 6 ⊢ Ⅎ𝑥(𝐵 ∖ (◡𝑅 “ 𝐴)) |
8 | 3, 7 | nfima 5393 | . . . . 5 ⊢ Ⅎ𝑥(𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))) |
9 | 6, 8 | nfun 3731 | . . . 4 ⊢ Ⅎ𝑥((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴)))) |
10 | 2, 9 | nfdif 3693 | . . 3 ⊢ Ⅎ𝑥(𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) |
11 | 10 | nfuni 4378 | . 2 ⊢ Ⅎ𝑥∪ (𝐵 ∖ ((◡𝑅 “ 𝐴) ∪ (𝑅 “ (𝐵 ∖ (◡𝑅 “ 𝐴))))) |
12 | 1, 11 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥sup(𝐴, 𝐵, 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2738 ∖ cdif 3537 ∪ cun 3538 ∪ cuni 4372 ◡ccnv 5037 “ cima 5041 supcsup 8229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-sup 8231 |
This theorem is referenced by: nfinf 8271 itg2cnlem1 23334 esum2d 29482 nfwlim 31012 totbndbnd 32758 aomclem8 36649 binomcxplemdvbinom 37574 binomcxplemdvsum 37576 binomcxplemnotnn0 37577 ssfiunibd 38464 fourierdlem20 39020 fourierdlem31 39031 fourierdlem79 39078 sge0ltfirp 39293 pimdecfgtioc 39602 decsmflem 39652 |
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