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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > watfvalN | Structured version Visualization version GIF version |
Description: 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 |
---|---|
watfvalN | β’ (πΎ β π΅ β π = (π β π΄ β¦ (π΄ β ((β₯πβπΎ)β{π})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3491 | . 2 β’ (πΎ β π΅ β πΎ β V) | |
2 | watomfval.w | . . 3 β’ π = (WAtomsβπΎ) | |
3 | fveq2 6878 | . . . . . 6 β’ (π = πΎ β (Atomsβπ) = (AtomsβπΎ)) | |
4 | watomfval.a | . . . . . 6 β’ π΄ = (AtomsβπΎ) | |
5 | 3, 4 | eqtr4di 2789 | . . . . 5 β’ (π = πΎ β (Atomsβπ) = π΄) |
6 | fveq2 6878 | . . . . . . 7 β’ (π = πΎ β (β₯πβπ) = (β₯πβπΎ)) | |
7 | 6 | fveq1d 6880 | . . . . . 6 β’ (π = πΎ β ((β₯πβπ)β{π}) = ((β₯πβπΎ)β{π})) |
8 | 5, 7 | difeq12d 4119 | . . . . 5 β’ (π = πΎ β ((Atomsβπ) β ((β₯πβπ)β{π})) = (π΄ β ((β₯πβπΎ)β{π}))) |
9 | 5, 8 | mpteq12dv 5232 | . . . 4 β’ (π = πΎ β (π β (Atomsβπ) β¦ ((Atomsβπ) β ((β₯πβπ)β{π}))) = (π β π΄ β¦ (π΄ β ((β₯πβπΎ)β{π})))) |
10 | df-watsN 38666 | . . . 4 β’ WAtoms = (π β V β¦ (π β (Atomsβπ) β¦ ((Atomsβπ) β ((β₯πβπ)β{π})))) | |
11 | 9, 10, 4 | mptfvmpt 7214 | . . 3 β’ (πΎ β V β (WAtomsβπΎ) = (π β π΄ β¦ (π΄ β ((β₯πβπΎ)β{π})))) |
12 | 2, 11 | eqtrid 2783 | . 2 β’ (πΎ β V β π = (π β π΄ β¦ (π΄ β ((β₯πβπΎ)β{π})))) |
13 | 1, 12 | syl 17 | 1 β’ (πΎ β π΅ β π = (π β π΄ β¦ (π΄ β ((β₯πβπΎ)β{π})))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1541 β wcel 2106 Vcvv 3473 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: watvalN 38669 |
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