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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 34110. (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 33909 | . . . 4 β’ (π β π β βͺ ran MblFnM) |
3 | 2 | mbfmfun 33905 | . . 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 33794 | . . . . . 6 β’ (π β (sigaGenβπ½) β βͺ ran sigAlgebra) |
7 | 4, 6, 1 | mbfmf 33906 | . . . . 5 β’ (π β π:βͺ πβΆβͺ (sigaGenβπ½)) |
8 | elex 3492 | . . . . . . 7 β’ (π½ β Top β π½ β V) | |
9 | unisg 33795 | . . . . . . 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 6714 | . . . . 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 6735 | . . 3 β’ (π β ran π β βͺ π½) |
14 | fimacnvinrn2 7087 | . . 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 582 | . 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 34110 | . 2 β’ (π β (πβRV/ππ π΄) = (β‘π β {π¦ β£ π¦π π΄})) |
18 | dfrab2 4313 | . . . 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 6068 | . 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 2778 | 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 2705 {crab 3430 Vcvv 3473 β© cin 3948 β wss 3949 βͺ cuni 4912 class class class wbr 5152 β‘ccnv 5681 ran crn 5683 β cima 5685 Fun wfun 6547 βΆwf 6549 βcfv 6553 (class class class)co 7426 Topctop 22815 sigAlgebracsiga 33760 sigaGencsigagen 33790 MblFnMcmbfm 33901 βRV/πcorvc 34108 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fo 6559 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 7999 df-2nd 8000 df-map 8853 df-siga 33761 df-sigagen 33791 df-mbfm 33902 df-orvc 34109 |
This theorem is referenced by: orvcoel 34114 orvccel 34115 orrvcval4 34117 |
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