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Mirrors > Home > MPE Home > Th. List > r1val3 | Structured version Visualization version GIF version |
Description: The value of the cumulative hierarchy of sets function expressed in terms of rank. Theorem 15.18 of [Monk1] p. 113. (Contributed by NM, 30-Nov-2003.) (Revised by Mario Carneiro, 17-Nov-2014.) |
Ref | Expression |
---|---|
r1val3 | β’ (π΄ β On β (π 1βπ΄) = βͺ π₯ β π΄ π« {π¦ β£ (rankβπ¦) β π₯}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1fnon 9744 | . . . . 5 β’ π 1 Fn On | |
2 | 1 | fndmi 6642 | . . . 4 β’ dom π 1 = On |
3 | 2 | eleq2i 2824 | . . 3 β’ (π΄ β dom π 1 β π΄ β On) |
4 | r1val1 9763 | . . 3 β’ (π΄ β dom π 1 β (π 1βπ΄) = βͺ π₯ β π΄ π« (π 1βπ₯)) | |
5 | 3, 4 | sylbir 234 | . 2 β’ (π΄ β On β (π 1βπ΄) = βͺ π₯ β π΄ π« (π 1βπ₯)) |
6 | onelon 6378 | . . . . 5 β’ ((π΄ β On β§ π₯ β π΄) β π₯ β On) | |
7 | r1val2 9814 | . . . . 5 β’ (π₯ β On β (π 1βπ₯) = {π¦ β£ (rankβπ¦) β π₯}) | |
8 | 6, 7 | syl 17 | . . . 4 β’ ((π΄ β On β§ π₯ β π΄) β (π 1βπ₯) = {π¦ β£ (rankβπ¦) β π₯}) |
9 | 8 | pweqd 4613 | . . 3 β’ ((π΄ β On β§ π₯ β π΄) β π« (π 1βπ₯) = π« {π¦ β£ (rankβπ¦) β π₯}) |
10 | 9 | iuneq2dv 5014 | . 2 β’ (π΄ β On β βͺ π₯ β π΄ π« (π 1βπ₯) = βͺ π₯ β π΄ π« {π¦ β£ (rankβπ¦) β π₯}) |
11 | 5, 10 | eqtrd 2771 | 1 β’ (π΄ β On β (π 1βπ΄) = βͺ π₯ β π΄ π« {π¦ β£ (rankβπ¦) β π₯}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 {cab 2708 π« cpw 4596 βͺ ciun 4990 dom cdm 5669 Oncon0 6353 βcfv 6532 π 1cr1 9739 rankcrnk 9740 |
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-pow 5356 ax-pr 5420 ax-un 7708 ax-reg 9569 ax-inf2 9618 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 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-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 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-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 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-ov 7396 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-r1 9741 df-rank 9742 |
This theorem is referenced by: (None) |
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