![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 6904 | . . . 4 β’ ((LTrnβπΎ)βπ) β V | |
2 | 1 | rabex 5328 | . . 3 β’ {π β ((LTrnβπΎ)βπ) β£ (((trLβπΎ)βπ)βπ) β€ π¦} β V |
3 | eqid 2728 | . . 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 6692 | . 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 2728 | . . . 4 β’ ((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) | |
9 | eqid 2728 | . . . 4 β’ ((trLβπΎ)βπ) = ((trLβπΎ)βπ) | |
10 | diafn.i | . . . 4 β’ πΌ = ((DIsoAβπΎ)βπ) | |
11 | 5, 6, 7, 8, 9, 10 | diafval 40498 | . . 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 6641 | . 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 395 = wceq 1534 β wcel 2099 {crab 3428 class class class wbr 5142 β¦ cmpt 5225 Fn wfn 6537 βcfv 6542 Basecbs 17173 lecple 17233 LHypclh 39451 LTrncltrn 39568 trLctrl 39625 DIsoAcdia 40495 |
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 5279 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
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 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-disoa 40496 |
This theorem is referenced by: diadm 40502 diaelrnN 40512 diaf11N 40516 |
Copyright terms: Public domain | W3C validator |