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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsfveq1 | Structured version Visualization version GIF version |
Description: If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-Jun-2021.) |
Ref | Expression |
---|---|
ntrcls.o | β’ π = (π β V β¦ (π β (π« π βm π« π) β¦ (π β π« π β¦ (π β (πβ(π β π)))))) |
ntrcls.d | β’ π· = (πβπ΅) |
ntrcls.r | β’ (π β πΌπ·πΎ) |
ntrclsfv.s | β’ (π β π β π« π΅) |
ntrclsfv.c | β’ (π β πΆ β π« π΅) |
Ref | Expression |
---|---|
ntrclsfveq1 | β’ (π β ((πΌβπ) = πΆ β (πΎβ(π΅ β π)) = (π΅ β πΆ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrclsfv.c | . . . . . 6 β’ (π β πΆ β π« π΅) | |
2 | 1 | elpwid 4610 | . . . . 5 β’ (π β πΆ β π΅) |
3 | dfss4 4257 | . . . . 5 β’ (πΆ β π΅ β (π΅ β (π΅ β πΆ)) = πΆ) | |
4 | 2, 3 | sylib 217 | . . . 4 β’ (π β (π΅ β (π΅ β πΆ)) = πΆ) |
5 | 4 | eqcomd 2736 | . . 3 β’ (π β πΆ = (π΅ β (π΅ β πΆ))) |
6 | 5 | eqeq2d 2741 | . 2 β’ (π β ((π΅ β (πΎβ(π΅ β π))) = πΆ β (π΅ β (πΎβ(π΅ β π))) = (π΅ β (π΅ β πΆ)))) |
7 | ntrcls.o | . . . 4 β’ π = (π β V β¦ (π β (π« π βm π« π) β¦ (π β π« π β¦ (π β (πβ(π β π)))))) | |
8 | ntrcls.d | . . . 4 β’ π· = (πβπ΅) | |
9 | ntrcls.r | . . . 4 β’ (π β πΌπ·πΎ) | |
10 | ntrclsfv.s | . . . 4 β’ (π β π β π« π΅) | |
11 | 7, 8, 9, 10 | ntrclsfv 43112 | . . 3 β’ (π β (πΌβπ) = (π΅ β (πΎβ(π΅ β π)))) |
12 | 11 | eqeq1d 2732 | . 2 β’ (π β ((πΌβπ) = πΆ β (π΅ β (πΎβ(π΅ β π))) = πΆ)) |
13 | 7, 8, 9 | ntrclskex 43107 | . . . . . 6 β’ (π β πΎ β (π« π΅ βm π« π΅)) |
14 | elmapi 8845 | . . . . . 6 β’ (πΎ β (π« π΅ βm π« π΅) β πΎ:π« π΅βΆπ« π΅) | |
15 | 13, 14 | syl 17 | . . . . 5 β’ (π β πΎ:π« π΅βΆπ« π΅) |
16 | 8, 9 | ntrclsrcomplex 43088 | . . . . 5 β’ (π β (π΅ β π) β π« π΅) |
17 | 15, 16 | ffvelcdmd 7086 | . . . 4 β’ (π β (πΎβ(π΅ β π)) β π« π΅) |
18 | 17 | elpwid 4610 | . . 3 β’ (π β (πΎβ(π΅ β π)) β π΅) |
19 | difssd 4131 | . . 3 β’ (π β (π΅ β πΆ) β π΅) | |
20 | rcompleq 4294 | . . 3 β’ (((πΎβ(π΅ β π)) β π΅ β§ (π΅ β πΆ) β π΅) β ((πΎβ(π΅ β π)) = (π΅ β πΆ) β (π΅ β (πΎβ(π΅ β π))) = (π΅ β (π΅ β πΆ)))) | |
21 | 18, 19, 20 | syl2anc 582 | . 2 β’ (π β ((πΎβ(π΅ β π)) = (π΅ β πΆ) β (π΅ β (πΎβ(π΅ β π))) = (π΅ β (π΅ β πΆ)))) |
22 | 6, 12, 21 | 3bitr4d 310 | 1 β’ (π β ((πΌβπ) = πΆ β (πΎβ(π΅ β π)) = (π΅ β πΆ))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1539 β wcel 2104 Vcvv 3472 β cdif 3944 β wss 3947 π« cpw 4601 class class class wbr 5147 β¦ cmpt 5230 βΆwf 6538 βcfv 6542 (class class class)co 7411 βm cmap 8822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-map 8824 |
This theorem is referenced by: ntrclsfveq 43115 |
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