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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 9669, which relies on rec, but theoretically 𝐶 in trcl 9669 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 9675 | . 2 ⊢ (∀𝑎(𝑎 ⊆ setrecs(𝐹) → 𝑎 ∈ setrecs(𝐹)) → setrecs(𝐹) = V) | |
2 | vex 3448 | . . . 4 ⊢ 𝑎 ∈ V | |
3 | 2 | pwid 4583 | . . 3 ⊢ 𝑎 ∈ 𝒫 𝑎 |
4 | pweq 4575 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → 𝒫 𝑥 = 𝒫 𝑎) | |
5 | vsetrec.1 | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ V ↦ 𝒫 𝑥) | |
6 | vpwex 5333 | . . . . . . 7 ⊢ 𝒫 𝑎 ∈ V | |
7 | 4, 5, 6 | fvmpt 6949 | . . . . . 6 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = 𝒫 𝑎) |
8 | 2, 7 | ax-mp 5 | . . . . 5 ⊢ (𝐹‘𝑎) = 𝒫 𝑎 |
9 | eqid 2733 | . . . . . 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 47222 | . . . . 5 ⊢ (𝑎 ⊆ setrecs(𝐹) → (𝐹‘𝑎) ⊆ setrecs(𝐹)) |
13 | 8, 12 | eqsstrrid 3994 | . . . 4 ⊢ (𝑎 ⊆ setrecs(𝐹) → 𝒫 𝑎 ⊆ setrecs(𝐹)) |
14 | 13 | sseld 3944 | . . 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 1800 | 1 ⊢ setrecs(𝐹) = V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ⊆ wss 3911 𝒫 cpw 4561 ↦ cmpt 5189 ‘cfv 6497 setrecscsetrecs 47214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-reg 9533 ax-inf2 9582 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-iin 4958 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-r1 9705 df-rank 9706 df-setrecs 47215 |
This theorem is referenced by: (None) |
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