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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dibclN | Structured version Visualization version GIF version |
Description: Closure of partial isomorphism B for a lattice πΎ. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dibcl.h | β’ π» = (LHypβπΎ) |
dibcl.i | β’ πΌ = ((DIsoBβπΎ)βπ) |
Ref | Expression |
---|---|
dibclN | β’ (((πΎ β HL β§ π β π») β§ π β dom πΌ) β (πΌβπ) β ran πΌ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dibcl.h | . . . 4 β’ π» = (LHypβπΎ) | |
2 | eqid 2728 | . . . 4 β’ ((DIsoAβπΎ)βπ) = ((DIsoAβπΎ)βπ) | |
3 | dibcl.i | . . . 4 β’ πΌ = ((DIsoBβπΎ)βπ) | |
4 | 1, 2, 3 | dibfna 40627 | . . 3 β’ ((πΎ β HL β§ π β π») β πΌ Fn dom ((DIsoAβπΎ)βπ)) |
5 | fnfun 6654 | . . 3 β’ (πΌ Fn dom ((DIsoAβπΎ)βπ) β Fun πΌ) | |
6 | 4, 5 | syl 17 | . 2 β’ ((πΎ β HL β§ π β π») β Fun πΌ) |
7 | fvelrn 7086 | . 2 β’ ((Fun πΌ β§ π β dom πΌ) β (πΌβπ) β ran πΌ) | |
8 | 6, 7 | sylan 579 | 1 β’ (((πΎ β HL β§ π β π») β§ π β dom πΌ) β (πΌβπ) β ran πΌ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 dom cdm 5678 ran crn 5679 Fun wfun 6542 Fn wfn 6543 βcfv 6548 HLchlt 38822 LHypclh 39457 DIsoAcdia 40501 DIsoBcdib 40611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-dib 40612 |
This theorem is referenced by: dibintclN 40640 |
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