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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 9746 | . . . . 5 β’ π 1 Fn On | |
2 | 1 | fndmi 6643 | . . . 4 β’ dom π 1 = On |
3 | 2 | eleq2i 2825 | . . 3 β’ (π΄ β dom π 1 β π΄ β On) |
4 | r1val1 9765 | . . 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 6379 | . . . . 5 β’ ((π΄ β On β§ π₯ β π΄) β π₯ β On) | |
7 | r1val2 9816 | . . . . 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 4614 | . . 3 β’ ((π΄ β On β§ π₯ β π΄) β π« (π 1βπ₯) = π« {π¦ β£ (rankβπ¦) β π₯}) |
10 | 9 | iuneq2dv 5015 | . 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 2772 | 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 2709 π« cpw 4597 βͺ ciun 4991 dom cdm 5670 Oncon0 6354 βcfv 6533 π 1cr1 9741 rankcrnk 9742 |
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 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-reg 9571 ax-inf2 9620 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7397 df-om 7840 df-2nd 7960 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-r1 9743 df-rank 9744 |
This theorem is referenced by: (None) |
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