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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvcval4 | Structured version Visualization version GIF version |
Description: The value of the preimage mapping operator can be restricted to preimages in the base set of the topology. Cf. orvcval 33985. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
Ref | Expression |
---|---|
orvccel.1 | β’ (π β π β βͺ ran sigAlgebra) |
orvccel.2 | β’ (π β π½ β Top) |
orvccel.3 | β’ (π β π β (πMblFnM(sigaGenβπ½))) |
orvccel.4 | β’ (π β π΄ β π) |
Ref | Expression |
---|---|
orvcval4 | β’ (π β (πβRV/ππ π΄) = (β‘π β {π¦ β βͺ π½ β£ π¦π π΄})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orvccel.3 | . . . . 5 β’ (π β π β (πMblFnM(sigaGenβπ½))) | |
2 | 1 | isanmbfm 33784 | . . . 4 β’ (π β π β βͺ ran MblFnM) |
3 | 2 | mbfmfun 33780 | . . 3 β’ (π β Fun π) |
4 | orvccel.1 | . . . . . 6 β’ (π β π β βͺ ran sigAlgebra) | |
5 | orvccel.2 | . . . . . . 7 β’ (π β π½ β Top) | |
6 | 5 | sgsiga 33669 | . . . . . 6 β’ (π β (sigaGenβπ½) β βͺ ran sigAlgebra) |
7 | 4, 6, 1 | mbfmf 33781 | . . . . 5 β’ (π β π:βͺ πβΆβͺ (sigaGenβπ½)) |
8 | elex 3487 | . . . . . . 7 β’ (π½ β Top β π½ β V) | |
9 | unisg 33670 | . . . . . . 7 β’ (π½ β V β βͺ (sigaGenβπ½) = βͺ π½) | |
10 | 5, 8, 9 | 3syl 18 | . . . . . 6 β’ (π β βͺ (sigaGenβπ½) = βͺ π½) |
11 | 10 | feq3d 6697 | . . . . 5 β’ (π β (π:βͺ πβΆβͺ (sigaGenβπ½) β π:βͺ πβΆβͺ π½)) |
12 | 7, 11 | mpbid 231 | . . . 4 β’ (π β π:βͺ πβΆβͺ π½) |
13 | 12 | frnd 6718 | . . 3 β’ (π β ran π β βͺ π½) |
14 | fimacnvinrn2 7067 | . . 3 β’ ((Fun π β§ ran π β βͺ π½) β (β‘π β {π¦ β£ π¦π π΄}) = (β‘π β ({π¦ β£ π¦π π΄} β© βͺ π½))) | |
15 | 3, 13, 14 | syl2anc 583 | . 2 β’ (π β (β‘π β {π¦ β£ π¦π π΄}) = (β‘π β ({π¦ β£ π¦π π΄} β© βͺ π½))) |
16 | orvccel.4 | . . 3 β’ (π β π΄ β π) | |
17 | 3, 1, 16 | orvcval 33985 | . 2 β’ (π β (πβRV/ππ π΄) = (β‘π β {π¦ β£ π¦π π΄})) |
18 | dfrab2 4305 | . . . 4 β’ {π¦ β βͺ π½ β£ π¦π π΄} = ({π¦ β£ π¦π π΄} β© βͺ π½) | |
19 | 18 | a1i 11 | . . 3 β’ (π β {π¦ β βͺ π½ β£ π¦π π΄} = ({π¦ β£ π¦π π΄} β© βͺ π½)) |
20 | 19 | imaeq2d 6052 | . 2 β’ (π β (β‘π β {π¦ β βͺ π½ β£ π¦π π΄}) = (β‘π β ({π¦ β£ π¦π π΄} β© βͺ π½))) |
21 | 15, 17, 20 | 3eqtr4d 2776 | 1 β’ (π β (πβRV/ππ π΄) = (β‘π β {π¦ β βͺ π½ β£ π¦π π΄})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 {cab 2703 {crab 3426 Vcvv 3468 β© cin 3942 β wss 3943 βͺ cuni 4902 class class class wbr 5141 β‘ccnv 5668 ran crn 5670 β cima 5672 Fun wfun 6530 βΆwf 6532 βcfv 6536 (class class class)co 7404 Topctop 22745 sigAlgebracsiga 33635 sigaGencsigagen 33665 MblFnMcmbfm 33776 βRV/πcorvc 33983 |
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 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
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 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fo 6542 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-map 8821 df-siga 33636 df-sigagen 33666 df-mbfm 33777 df-orvc 33984 |
This theorem is referenced by: orvcoel 33989 orvccel 33990 orrvcval4 33992 |
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