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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrelmap | Structured version Visualization version GIF version |
Description: The interior function is a map from the powerset of the base set to itself. (Contributed by RP, 22-Apr-2021.) |
Ref | Expression |
---|---|
ntrrn.x | β’ π = βͺ π½ |
ntrrn.i | β’ πΌ = (intβπ½) |
Ref | Expression |
---|---|
ntrelmap | β’ (π½ β Top β πΌ β (π« π βm π« π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrrn.x | . . 3 β’ π = βͺ π½ | |
2 | ntrrn.i | . . 3 β’ πΌ = (intβπ½) | |
3 | 1, 2 | ntrf2 42063 | . 2 β’ (π½ β Top β πΌ:π« πβΆπ« π) |
4 | 1 | topopn 22161 | . . . 4 β’ (π½ β Top β π β π½) |
5 | 4 | pwexd 5322 | . . 3 β’ (π½ β Top β π« π β V) |
6 | 5, 5 | elmapd 8700 | . 2 β’ (π½ β Top β (πΌ β (π« π βm π« π) β πΌ:π« πβΆπ« π)) |
7 | 3, 6 | mpbird 256 | 1 β’ (π½ β Top β πΌ β (π« π βm π« π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 Vcvv 3441 π« cpw 4547 βͺ cuni 4852 βΆwf 6475 βcfv 6479 (class class class)co 7337 βm cmap 8686 Topctop 22148 intcnt 22274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-map 8688 df-top 22149 df-topon 22166 df-ntr 22277 |
This theorem is referenced by: (None) |
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