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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > watvalN | Structured version Visualization version GIF version |
Description: Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
watomfval.a | β’ π΄ = (AtomsβπΎ) |
watomfval.p | β’ π = (β₯πβπΎ) |
watomfval.w | β’ π = (WAtomsβπΎ) |
Ref | Expression |
---|---|
watvalN | β’ ((πΎ β π΅ β§ π· β π΄) β (πβπ·) = (π΄ β ((β₯πβπΎ)β{π·}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | watomfval.a | . . . 4 β’ π΄ = (AtomsβπΎ) | |
2 | watomfval.p | . . . 4 β’ π = (β₯πβπΎ) | |
3 | watomfval.w | . . . 4 β’ π = (WAtomsβπΎ) | |
4 | 1, 2, 3 | watfvalN 38668 | . . 3 β’ (πΎ β π΅ β π = (π β π΄ β¦ (π΄ β ((β₯πβπΎ)β{π})))) |
5 | 4 | fveq1d 6880 | . 2 β’ (πΎ β π΅ β (πβπ·) = ((π β π΄ β¦ (π΄ β ((β₯πβπΎ)β{π})))βπ·)) |
6 | sneq 4632 | . . . . 5 β’ (π = π· β {π} = {π·}) | |
7 | 6 | fveq2d 6882 | . . . 4 β’ (π = π· β ((β₯πβπΎ)β{π}) = ((β₯πβπΎ)β{π·})) |
8 | 7 | difeq2d 4118 | . . 3 β’ (π = π· β (π΄ β ((β₯πβπΎ)β{π})) = (π΄ β ((β₯πβπΎ)β{π·}))) |
9 | eqid 2731 | . . 3 β’ (π β π΄ β¦ (π΄ β ((β₯πβπΎ)β{π}))) = (π β π΄ β¦ (π΄ β ((β₯πβπΎ)β{π}))) | |
10 | 1 | fvexi 6892 | . . . 4 β’ π΄ β V |
11 | 10 | difexi 5321 | . . 3 β’ (π΄ β ((β₯πβπΎ)β{π·})) β V |
12 | 8, 9, 11 | fvmpt 6984 | . 2 β’ (π· β π΄ β ((π β π΄ β¦ (π΄ β ((β₯πβπΎ)β{π})))βπ·) = (π΄ β ((β₯πβπΎ)β{π·}))) |
13 | 5, 12 | sylan9eq 2791 | 1 β’ ((πΎ β π΅ β§ π· β π΄) β (πβπ·) = (π΄ β ((β₯πβπΎ)β{π·}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 β cdif 3941 {csn 4622 β¦ cmpt 5224 βcfv 6532 Atomscatm 37938 β₯πcpolN 38578 WAtomscwpointsN 38662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-watsN 38666 |
This theorem is referenced by: iswatN 38670 |
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