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Mirrors > Home > MPE Home > Th. List > r1suc | Structured version Visualization version GIF version |
Description: Value of the cumulative hierarchy of sets function at a successor ordinal. Part of Definition 9.9 of [TakeutiZaring] p. 76. (Contributed by NM, 2-Sep-2003.) (Revised by Mario Carneiro, 10-Sep-2013.) |
Ref | Expression |
---|---|
r1suc | ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r1sucg 9527 | . 2 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | |
2 | r1fnon 9525 | . . . 4 ⊢ 𝑅1 Fn On | |
3 | 2 | fndmi 6537 | . . 3 ⊢ dom 𝑅1 = On |
4 | 3 | eqcomi 2747 | . 2 ⊢ On = dom 𝑅1 |
5 | 1, 4 | eleq2s 2857 | 1 ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 𝒫 cpw 4533 dom cdm 5589 Oncon0 6266 suc csuc 6268 ‘cfv 6433 𝑅1cr1 9520 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-r1 9522 |
This theorem is referenced by: r1sdom 9532 r1sssuc 9541 tz9.12lem3 9547 rankval2 9576 rankpwi 9581 dfac12lem2 9900 dfac12r 9902 ackbij2lem2 9996 ackbij2lem3 9997 wunr1om 10475 r1wunlim 10493 tskr1om 10523 inar1 10531 inatsk 10534 grur1a 10575 grothomex 10585 rankeq1o 34473 elhf2 34477 0hf 34479 aomclem1 40879 grur1cld 41850 |
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