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Mirrors > Home > MPE Home > Th. List > Mathboxes > diafn | Structured version Visualization version GIF version |
Description: Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.) |
Ref | Expression |
---|---|
diafn.b | β’ π΅ = (BaseβπΎ) |
diafn.l | β’ β€ = (leβπΎ) |
diafn.h | β’ π» = (LHypβπΎ) |
diafn.i | β’ πΌ = ((DIsoAβπΎ)βπ) |
Ref | Expression |
---|---|
diafn | β’ ((πΎ β π β§ π β π») β πΌ Fn {π₯ β π΅ β£ π₯ β€ π}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6859 | . . . 4 β’ ((LTrnβπΎ)βπ) β V | |
2 | 1 | rabex 5293 | . . 3 β’ {π β ((LTrnβπΎ)βπ) β£ (((trLβπΎ)βπ)βπ) β€ π¦} β V |
3 | eqid 2733 | . . 3 β’ (π¦ β {π₯ β π΅ β£ π₯ β€ π} β¦ {π β ((LTrnβπΎ)βπ) β£ (((trLβπΎ)βπ)βπ) β€ π¦}) = (π¦ β {π₯ β π΅ β£ π₯ β€ π} β¦ {π β ((LTrnβπΎ)βπ) β£ (((trLβπΎ)βπ)βπ) β€ π¦}) | |
4 | 2, 3 | fnmpti 6648 | . 2 β’ (π¦ β {π₯ β π΅ β£ π₯ β€ π} β¦ {π β ((LTrnβπΎ)βπ) β£ (((trLβπΎ)βπ)βπ) β€ π¦}) Fn {π₯ β π΅ β£ π₯ β€ π} |
5 | diafn.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
6 | diafn.l | . . . 4 β’ β€ = (leβπΎ) | |
7 | diafn.h | . . . 4 β’ π» = (LHypβπΎ) | |
8 | eqid 2733 | . . . 4 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
9 | eqid 2733 | . . . 4 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
10 | diafn.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
11 | 5, 6, 7, 8, 9, 10 | diafval 39544 | . . 3 β’ ((πΎ β π β§ π β π») β πΌ = (π¦ β {π₯ β π΅ β£ π₯ β€ π} β¦ {π β ((LTrnβπΎ)βπ) β£ (((trLβπΎ)βπ)βπ) β€ π¦})) |
12 | 11 | fneq1d 6599 | . 2 β’ ((πΎ β π β§ π β π») β (πΌ Fn {π₯ β π΅ β£ π₯ β€ π} β (π¦ β {π₯ β π΅ β£ π₯ β€ π} β¦ {π β ((LTrnβπΎ)βπ) β£ (((trLβπΎ)βπ)βπ) β€ π¦}) Fn {π₯ β π΅ β£ π₯ β€ π})) |
13 | 4, 12 | mpbiri 258 | 1 β’ ((πΎ β π β§ π β π») β πΌ Fn {π₯ β π΅ β£ π₯ β€ π}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3406 class class class wbr 5109 β¦ cmpt 5192 Fn wfn 6495 βcfv 6500 Basecbs 17091 lecple 17148 LHypclh 38497 LTrncltrn 38614 trLctrl 38671 DIsoAcdia 39541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-disoa 39542 |
This theorem is referenced by: diadm 39548 diaelrnN 39558 diaf11N 39562 |
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