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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 9186 | . 2 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | |
2 | r1fnon 9184 | . . . 4 ⊢ 𝑅1 Fn On | |
3 | fndm 6448 | . . . 4 ⊢ (𝑅1 Fn On → dom 𝑅1 = On) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ dom 𝑅1 = On |
5 | 4 | eqcomi 2827 | . 2 ⊢ On = dom 𝑅1 |
6 | 1, 5 | eleq2s 2928 | 1 ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 𝒫 cpw 4535 dom cdm 5548 Oncon0 6184 suc csuc 6186 Fn wfn 6343 ‘cfv 6348 𝑅1cr1 9179 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-r1 9181 |
This theorem is referenced by: r1sdom 9191 r1sssuc 9200 tz9.12lem3 9206 rankval2 9235 rankpwi 9240 dfac12lem2 9558 dfac12r 9560 ackbij2lem2 9650 ackbij2lem3 9651 wunr1om 10129 r1wunlim 10147 tskr1om 10177 inar1 10185 inatsk 10188 grur1a 10229 grothomex 10239 rankeq1o 33529 elhf2 33533 0hf 33535 aomclem1 39532 grur1cld 40445 |
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