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Mirrors > Home > MPE Home > Th. List > Mathboxes > vsetrec | Structured version Visualization version GIF version |
Description: Construct V using set recursion. The proof indirectly uses trcl 8604, which relies on rec, but theoretically 𝐶 in trcl 8604 could be constructed using setrecs instead. The proof of this theorem uses the dummy variable 𝑎 rather than 𝑥 to avoid a distinct variable requirement between 𝐹 and 𝑥. (Contributed by Emmett Weisz, 23-Jun-2021.) |
Ref | Expression |
---|---|
vsetrec.1 | ⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥) |
Ref | Expression |
---|---|
vsetrec | ⊢ setrecs(𝐹) = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setind 8610 | . 2 ⊢ (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V) | |
2 | vex 3203 | . . . 4 ⊢ 𝑎 ∈ V | |
3 | 2 | pwid 4174 | . . 3 ⊢ 𝑎 ∈ 𝒫 𝑎 |
4 | pweq 4161 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎) | |
5 | vsetrec.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥) | |
6 | 2 | pwex 4848 | . . . . . . 7 ⊢ 𝒫 𝑎 ∈ V |
7 | 4, 5, 6 | fvmpt 6282 | . . . . . 6 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = 𝒫 𝑎) |
8 | 2, 7 | ax-mp 5 | . . . . 5 ⊢ (𝐹‘𝑎) = 𝒫 𝑎 |
9 | eqid 2622 | . . . . . 6 ⊢ setrecs(𝐹) = setrecs(𝐹) | |
10 | 2 | a1i 11 | . . . . . 6 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ V) |
11 | id 22 | . . . . . 6 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ⊆ setrecs(𝐹)) | |
12 | 9, 10, 11 | setrec1 42438 | . . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹)) |
13 | 8, 12 | syl5eqssr 3650 | . . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹)) |
14 | 13 | sseld 3602 | . . 3 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝑎 ∈ 𝒫 𝑎 → 𝑎 ∈ setrecs(𝐹))) |
15 | 3, 14 | mpi 20 | . 2 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) |
16 | 1, 15 | mpg 1724 | 1 ⊢ setrecs(𝐹) = V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 𝒫 cpw 4158 ↦ cmpt 4729 ‘cfv 5888 setrecscsetrecs 42430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-reg 8497 ax-inf2 8538 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-r1 8627 df-rank 8628 df-setrecs 42431 |
This theorem is referenced by: (None) |
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