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Mathbox for Norm Megill |
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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 7336 | . 2 β’ (β!π₯ β π΄ π β (β©π₯ β π΄ π) β π΄) | |
2 | undefnel2 8213 | . . . 4 β’ (π΄ β π β Β¬ (Undefβπ΄) β π΄) | |
3 | iffalse 4500 | . . . . . . 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 37444 | . . . . . . 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 2875 | . . . . . . . 8 β’ π΄ = {π₯ β£ π₯ β π΄} | |
6 | 5 | fveq2i 6850 | . . . . . . 7 β’ (Undefβπ΄) = (Undefβ{π₯ β£ π₯ β π΄}) |
7 | 3, 4, 6 | 3eqtr4g 2802 | . . . . . 6 β’ (Β¬ β!π₯ β π΄ π β (β©π₯ β π΄ π) = (Undefβπ΄)) |
8 | 7 | eleq1d 2823 | . . . . 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 247 | . . 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 397 β wcel 2107 {cab 2714 β!wreu 3354 ifcif 4491 β©cio 6451 βcfv 6501 β©crio 7317 Undefcund 8208 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-riotaBAD 37444 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-mpt 5194 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-iota 6453 df-fun 6503 df-fv 6509 df-riota 7318 df-undef 8209 |
This theorem is referenced by: riotaclbBAD 37446 riotasvd 37447 |
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