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Mirrors > Home > MPE Home > Th. List > r1sucg | 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 Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
r1sucg | ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgsucg 8061 | . . 3 ⊢ (𝐴 ∈ dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) → (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴))) | |
2 | df-r1 9195 | . . . 4 ⊢ 𝑅1 = rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) | |
3 | 2 | dmeqi 5775 | . . 3 ⊢ dom 𝑅1 = dom rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅) |
4 | 1, 3 | eleq2s 2933 | . 2 ⊢ (𝐴 ∈ dom 𝑅1 → (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴))) |
5 | 2 | fveq1i 6673 | . 2 ⊢ (𝑅1‘suc 𝐴) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘suc 𝐴) |
6 | fvex 6685 | . . . 4 ⊢ (𝑅1‘𝐴) ∈ V | |
7 | pweq 4557 | . . . . 5 ⊢ (𝑥 = (𝑅1‘𝐴) → 𝒫 𝑥 = 𝒫 (𝑅1‘𝐴)) | |
8 | eqid 2823 | . . . . 5 ⊢ (𝑥 ∈ V ↦ 𝒫 𝑥) = (𝑥 ∈ V ↦ 𝒫 𝑥) | |
9 | 6 | pwex 5283 | . . . . 5 ⊢ 𝒫 (𝑅1‘𝐴) ∈ V |
10 | 7, 8, 9 | fvmpt 6770 | . . . 4 ⊢ ((𝑅1‘𝐴) ∈ V → ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1‘𝐴)) = 𝒫 (𝑅1‘𝐴)) |
11 | 6, 10 | ax-mp 5 | . . 3 ⊢ ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1‘𝐴)) = 𝒫 (𝑅1‘𝐴) |
12 | 2 | fveq1i 6673 | . . . 4 ⊢ (𝑅1‘𝐴) = (rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴) |
13 | 12 | fveq2i 6675 | . . 3 ⊢ ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(𝑅1‘𝐴)) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)) |
14 | 11, 13 | eqtr3i 2848 | . 2 ⊢ 𝒫 (𝑅1‘𝐴) = ((𝑥 ∈ V ↦ 𝒫 𝑥)‘(rec((𝑥 ∈ V ↦ 𝒫 𝑥), ∅)‘𝐴)) |
15 | 4, 5, 14 | 3eqtr4g 2883 | 1 ⊢ (𝐴 ∈ dom 𝑅1 → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∅c0 4293 𝒫 cpw 4541 ↦ cmpt 5148 dom cdm 5557 suc csuc 6195 ‘cfv 6357 reccrdg 8047 𝑅1cr1 9193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-r1 9195 |
This theorem is referenced by: r1suc 9201 r1fin 9204 r1tr 9207 r1ordg 9209 r1pwss 9215 r1val1 9217 rankwflemb 9224 r1elwf 9227 rankr1ai 9229 rankr1bg 9234 pwwf 9238 unwf 9241 uniwf 9250 rankonidlem 9259 rankr1id 9293 |
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