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Mathbox for Richard Penner |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsfveq | Structured version Visualization version GIF version |
Description: If interior and closure functions are related then equality of a pair of function values is equivalent to equality of a pair of the other function's values. (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.t | β’ (π β π β π« π΅) |
Ref | Expression |
---|---|
ntrclsfveq | β’ (π β ((πΌβπ) = (πΌβπ) β (πΎβ(π΅ β π)) = (πΎβ(π΅ β π)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 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)))))) | |
2 | ntrcls.d | . . . 4 β’ π· = (πβπ΅) | |
3 | ntrcls.r | . . . 4 β’ (π β πΌπ·πΎ) | |
4 | ntrclsfv.t | . . . 4 β’ (π β π β π« π΅) | |
5 | 1, 2, 3, 4 | ntrclsfv 43299 | . . 3 β’ (π β (πΌβπ) = (π΅ β (πΎβ(π΅ β π)))) |
6 | 5 | eqeq2d 2735 | . 2 β’ (π β ((πΌβπ) = (πΌβπ) β (πΌβπ) = (π΅ β (πΎβ(π΅ β π))))) |
7 | ntrclsfv.s | . . 3 β’ (π β π β π« π΅) | |
8 | 2, 3 | ntrclsrcomplex 43275 | . . 3 β’ (π β (π΅ β (πΎβ(π΅ β π))) β π« π΅) |
9 | 1, 2, 3, 7, 8 | ntrclsfveq1 43300 | . 2 β’ (π β ((πΌβπ) = (π΅ β (πΎβ(π΅ β π))) β (πΎβ(π΅ β π)) = (π΅ β (π΅ β (πΎβ(π΅ β π)))))) |
10 | 1, 2, 3 | ntrclskex 43294 | . . . . . . 7 β’ (π β πΎ β (π« π΅ βm π« π΅)) |
11 | elmapi 8839 | . . . . . . 7 β’ (πΎ β (π« π΅ βm π« π΅) β πΎ:π« π΅βΆπ« π΅) | |
12 | 10, 11 | syl 17 | . . . . . 6 β’ (π β πΎ:π« π΅βΆπ« π΅) |
13 | 2, 3 | ntrclsrcomplex 43275 | . . . . . 6 β’ (π β (π΅ β π) β π« π΅) |
14 | 12, 13 | ffvelcdmd 7077 | . . . . 5 β’ (π β (πΎβ(π΅ β π)) β π« π΅) |
15 | 14 | elpwid 4603 | . . . 4 β’ (π β (πΎβ(π΅ β π)) β π΅) |
16 | dfss4 4250 | . . . 4 β’ ((πΎβ(π΅ β π)) β π΅ β (π΅ β (π΅ β (πΎβ(π΅ β π)))) = (πΎβ(π΅ β π))) | |
17 | 15, 16 | sylib 217 | . . 3 β’ (π β (π΅ β (π΅ β (πΎβ(π΅ β π)))) = (πΎβ(π΅ β π))) |
18 | 17 | eqeq2d 2735 | . 2 β’ (π β ((πΎβ(π΅ β π)) = (π΅ β (π΅ β (πΎβ(π΅ β π)))) β (πΎβ(π΅ β π)) = (πΎβ(π΅ β π)))) |
19 | 6, 9, 18 | 3bitrd 305 | 1 β’ (π β ((πΌβπ) = (πΌβπ) β (πΎβ(π΅ β π)) = (πΎβ(π΅ β π)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1533 β wcel 2098 Vcvv 3466 β cdif 3937 β wss 3940 π« cpw 4594 class class class wbr 5138 β¦ cmpt 5221 βΆwf 6529 βcfv 6533 (class class class)co 7401 βm cmap 8816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-1st 7968 df-2nd 7969 df-map 8818 |
This theorem is referenced by: (None) |
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