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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 9718, which relies on rec, but theoretically 𝐶 in trcl 9718 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 9724 | . 2 ⊢ (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V) | |
2 | vex 3470 | . . . 4 ⊢ 𝑎 ∈ V | |
3 | 2 | pwid 4616 | . . 3 ⊢ 𝑎 ∈ 𝒫 𝑎 |
4 | pweq 4608 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎) | |
5 | vsetrec.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥) | |
6 | vpwex 5365 | . . . . . . 7 ⊢ 𝒫 𝑎 ∈ V | |
7 | 4, 5, 6 | fvmpt 6988 | . . . . . 6 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = 𝒫 𝑎) |
8 | 2, 7 | ax-mp 5 | . . . . 5 ⊢ (𝐹‘𝑎) = 𝒫 𝑎 |
9 | eqid 2724 | . . . . . 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 47890 | . . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹)) |
13 | 8, 12 | eqsstrrid 4023 | . . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹)) |
14 | 13 | sseld 3973 | . . 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 3466 ⊆ wss 3940 𝒫 cpw 4594 ↦ cmpt 5221 ‘cfv 6533 setrecscsetrecs 47882 |
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 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-reg 9582 ax-inf2 9631 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-r1 9754 df-rank 9755 df-setrecs 47883 |
This theorem is referenced by: (None) |
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