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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 9759, which relies on rec, but theoretically 𝐶 in trcl 9759 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 9765 | . 2 ⊢ (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V) | |
2 | vex 3477 | . . . 4 ⊢ 𝑎 ∈ V | |
3 | 2 | pwid 4628 | . . 3 ⊢ 𝑎 ∈ 𝒫 𝑎 |
4 | pweq 4620 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎) | |
5 | vsetrec.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥) | |
6 | vpwex 5381 | . . . . . . 7 ⊢ 𝒫 𝑎 ∈ V | |
7 | 4, 5, 6 | fvmpt 7010 | . . . . . 6 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = 𝒫 𝑎) |
8 | 2, 7 | ax-mp 5 | . . . . 5 ⊢ (𝐹‘𝑎) = 𝒫 𝑎 |
9 | eqid 2728 | . . . . . 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 48200 | . . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹)) |
13 | 8, 12 | eqsstrrid 4031 | . . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹)) |
14 | 13 | sseld 3981 | . . 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 1791 | 1 ⊢ setrecs(𝐹) = V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 Vcvv 3473 ⊆ wss 3949 𝒫 cpw 4606 ↦ cmpt 5235 ‘cfv 6553 setrecscsetrecs 48192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-reg 9623 ax-inf2 9672 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-iin 5003 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-r1 9795 df-rank 9796 df-setrecs 48193 |
This theorem is referenced by: (None) |
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