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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 9283 | . 2 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) | |
2 | r1fnon 9281 | . . . 4 ⊢ 𝑅1 Fn On | |
3 | 2 | fndmi 6451 | . . 3 ⊢ dom 𝑅1 = On |
4 | 3 | eqcomi 2748 | . 2 ⊢ On = dom 𝑅1 |
5 | 1, 4 | eleq2s 2852 | 1 ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 𝒫 cpw 4498 dom cdm 5535 Oncon0 6182 suc csuc 6184 ‘cfv 6349 𝑅1cr1 9276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-r1 9278 |
This theorem is referenced by: r1sdom 9288 r1sssuc 9297 tz9.12lem3 9303 rankval2 9332 rankpwi 9337 dfac12lem2 9656 dfac12r 9658 ackbij2lem2 9752 ackbij2lem3 9753 wunr1om 10231 r1wunlim 10249 tskr1om 10279 inar1 10287 inatsk 10290 grur1a 10331 grothomex 10341 rankeq1o 34128 elhf2 34132 0hf 34134 aomclem1 40491 grur1cld 41432 |
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