![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > riotaclbgBAD | Structured version Visualization version GIF version |
Description: Closure of restricted iota. (Contributed by NM, 28-Feb-2013.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
riotaclbgBAD | β’ (π΄ β π β (β!π₯ β π΄ π β (β©π₯ β π΄ π) β π΄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | riotacl 7388 | . 2 β’ (β!π₯ β π΄ π β (β©π₯ β π΄ π) β π΄) | |
2 | undefnel2 8274 | . . . 4 β’ (π΄ β π β Β¬ (Undefβπ΄) β π΄) | |
3 | iffalse 4533 | . . . . . . 7 β’ (Β¬ β!π₯ β π΄ π β if(β!π₯ β π΄ π, (β©π₯(π₯ β π΄ β§ π)), (Undefβ{π₯ β£ π₯ β π΄})) = (Undefβ{π₯ β£ π₯ β π΄})) | |
4 | ax-riotaBAD 38349 | . . . . . . 7 β’ (β©π₯ β π΄ π) = if(β!π₯ β π΄ π, (β©π₯(π₯ β π΄ β§ π)), (Undefβ{π₯ β£ π₯ β π΄})) | |
5 | abid1 2865 | . . . . . . . 8 β’ π΄ = {π₯ β£ π₯ β π΄} | |
6 | 5 | fveq2i 6894 | . . . . . . 7 β’ (Undefβπ΄) = (Undefβ{π₯ β£ π₯ β π΄}) |
7 | 3, 4, 6 | 3eqtr4g 2792 | . . . . . 6 β’ (Β¬ β!π₯ β π΄ π β (β©π₯ β π΄ π) = (Undefβπ΄)) |
8 | 7 | eleq1d 2813 | . . . . 5 β’ (Β¬ β!π₯ β π΄ π β ((β©π₯ β π΄ π) β π΄ β (Undefβπ΄) β π΄)) |
9 | 8 | notbid 318 | . . . 4 β’ (Β¬ β!π₯ β π΄ π β (Β¬ (β©π₯ β π΄ π) β π΄ β Β¬ (Undefβπ΄) β π΄)) |
10 | 2, 9 | syl5ibrcom 246 | . . 3 β’ (π΄ β π β (Β¬ β!π₯ β π΄ π β Β¬ (β©π₯ β π΄ π) β π΄)) |
11 | 10 | con4d 115 | . 2 β’ (π΄ β π β ((β©π₯ β π΄ π) β π΄ β β!π₯ β π΄ π)) |
12 | 1, 11 | impbid2 225 | 1 β’ (π΄ β π β (β!π₯ β π΄ π β (β©π₯ β π΄ π) β π΄)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β wb 205 β§ wa 395 β wcel 2099 {cab 2704 β!wreu 3369 ifcif 4524 β©cio 6492 βcfv 6542 β©crio 7369 Undefcund 8269 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-riotaBAD 38349 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 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-iota 6494 df-fun 6544 df-fv 6550 df-riota 7370 df-undef 8270 |
This theorem is referenced by: riotaclbBAD 38351 riotasvd 38352 |
Copyright terms: Public domain | W3C validator |