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Mirrors > Home > MPE Home > Th. List > rankvalb | Structured version Visualization version GIF version |
Description: Value of the rank function. Definition 9.14 of [TakeutiZaring] p. 79 (proved as a theorem from our definition). This variant of rankval 9837 does not use Regularity, and so requires the assumption that π΄ is in the range of π 1. (Contributed by NM, 11-Oct-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
Ref | Expression |
---|---|
rankvalb | β’ (π΄ β βͺ (π 1 β On) β (rankβπ΄) = β© {π₯ β On β£ π΄ β (π 1βsuc π₯)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rank 9786 | . 2 β’ rank = (π¦ β V β¦ β© {π₯ β On β£ π¦ β (π 1βsuc π₯)}) | |
2 | eleq1 2813 | . . . 4 β’ (π¦ = π΄ β (π¦ β (π 1βsuc π₯) β π΄ β (π 1βsuc π₯))) | |
3 | 2 | rabbidv 3427 | . . 3 β’ (π¦ = π΄ β {π₯ β On β£ π¦ β (π 1βsuc π₯)} = {π₯ β On β£ π΄ β (π 1βsuc π₯)}) |
4 | 3 | inteqd 4947 | . 2 β’ (π¦ = π΄ β β© {π₯ β On β£ π¦ β (π 1βsuc π₯)} = β© {π₯ β On β£ π΄ β (π 1βsuc π₯)}) |
5 | elex 3482 | . 2 β’ (π΄ β βͺ (π 1 β On) β π΄ β V) | |
6 | rankwflemb 9814 | . . 3 β’ (π΄ β βͺ (π 1 β On) β βπ₯ β On π΄ β (π 1βsuc π₯)) | |
7 | intexrab 5335 | . . 3 β’ (βπ₯ β On π΄ β (π 1βsuc π₯) β β© {π₯ β On β£ π΄ β (π 1βsuc π₯)} β V) | |
8 | 6, 7 | sylbb 218 | . 2 β’ (π΄ β βͺ (π 1 β On) β β© {π₯ β On β£ π΄ β (π 1βsuc π₯)} β V) |
9 | 1, 4, 5, 8 | fvmptd3 7021 | 1 β’ (π΄ β βͺ (π 1 β On) β (rankβπ΄) = β© {π₯ β On β£ π΄ β (π 1βsuc π₯)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 βwrex 3060 {crab 3419 Vcvv 3463 βͺ cuni 4901 β© cint 4942 β cima 5673 Oncon0 6362 suc csuc 6364 βcfv 6541 π 1cr1 9783 rankcrnk 9784 |
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 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4943 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7417 df-om 7867 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-r1 9785 df-rank 9786 |
This theorem is referenced by: rankr1ai 9819 rankidb 9821 rankval 9837 |
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