![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 9743 | . . . . 5 β’ π 1 Fn On | |
2 | 1 | fndmi 6641 | . . . 4 β’ dom π 1 = On |
3 | 2 | eleq2i 2824 | . . 3 β’ (π΄ β dom π 1 β π΄ β On) |
4 | r1val1 9762 | . . 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 6377 | . . . . 5 β’ ((π΄ β On β§ π₯ β π΄) β π₯ β On) | |
7 | r1val2 9813 | . . . . 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 4612 | . . 3 β’ ((π΄ β On β§ π₯ β π΄) β π« (π 1βπ₯) = π« {π¦ β£ (rankβπ¦) β π₯}) |
10 | 9 | iuneq2dv 5013 | . 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 4595 βͺ ciun 4989 dom cdm 5668 Oncon0 6352 βcfv 6531 π 1cr1 9738 rankcrnk 9739 |
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 5277 ax-sep 5291 ax-nul 5298 ax-pow 5355 ax-pr 5419 ax-un 7707 ax-reg 9568 ax-inf2 9617 |
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 3474 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-int 4943 df-iun 4991 df-br 5141 df-opab 5203 df-mpt 5224 df-tr 5258 df-id 5566 df-eprel 5572 df-po 5580 df-so 5581 df-fr 5623 df-we 5625 df-xp 5674 df-rel 5675 df-cnv 5676 df-co 5677 df-dm 5678 df-rn 5679 df-res 5680 df-ima 5681 df-pred 6288 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7395 df-om 7838 df-2nd 7957 df-frecs 8247 df-wrecs 8278 df-recs 8352 df-rdg 8391 df-r1 9740 df-rank 9741 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |