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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 7387 | . 2 β’ (β!π₯ β π΄ π β (β©π₯ β π΄ π) β π΄) | |
2 | undefnel2 8276 | . . . 4 β’ (π΄ β π β Β¬ (Undefβπ΄) β π΄) | |
3 | iffalse 4534 | . . . . . . 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 38477 | . . . . . . 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 2862 | . . . . . . . 8 β’ π΄ = {π₯ β£ π₯ β π΄} | |
6 | 5 | fveq2i 6893 | . . . . . . 7 β’ (Undefβπ΄) = (Undefβ{π₯ β£ π₯ β π΄}) |
7 | 3, 4, 6 | 3eqtr4g 2790 | . . . . . 6 β’ (Β¬ β!π₯ β π΄ π β (β©π₯ β π΄ π) = (Undefβπ΄)) |
8 | 7 | eleq1d 2810 | . . . . 5 β’ (Β¬ β!π₯ β π΄ π β ((β©π₯ β π΄ π) β π΄ β (Undefβπ΄) β π΄)) |
9 | 8 | notbid 317 | . . . 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 394 β wcel 2098 {cab 2702 β!wreu 3362 ifcif 4525 β©cio 6493 βcfv 6543 β©crio 7368 Undefcund 8271 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-riotaBAD 38477 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-iota 6495 df-fun 6545 df-fv 6551 df-riota 7369 df-undef 8272 |
This theorem is referenced by: riotaclbBAD 38479 riotasvd 38480 |
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