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Theorem setrec2 49214
Description: This is the second of two fundamental theorems about set recursion from which all other facts will be derived. It states that the class setrecs(𝐹) is a subclass of all classes 𝐶 that are closed under 𝐹. Taken together, Theorems setrec1 49210 and setrec2v 49215 uniquely determine setrecs(𝐹) to be the minimal class closed under 𝐹.

We express this by saying that if 𝐹 respects the relation and 𝐶 is closed under 𝐹, then 𝐵𝐶. By substituting strategically constructed classes for 𝐶, we can easily prove many useful properties. Although this theorem cannot show equality between 𝐵 and 𝐶, if we intend to prove equality between 𝐵 and some particular class (such as On), we first apply this theorem, then the relevant induction theorem (such as tfi 7874) to the other class.

(Contributed by Emmett Weisz, 2-Sep-2021.)

Hypotheses
Ref Expression
setrec2.1 𝑎𝐹
setrec2.2 𝐵 = setrecs(𝐹)
setrec2.3 (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
Assertion
Ref Expression
setrec2 (𝜑𝐵𝐶)
Distinct variable group:   𝐶,𝑎
Allowed substitution hints:   𝜑(𝑎)   𝐵(𝑎)   𝐹(𝑎)

Proof of Theorem setrec2
Dummy variables 𝑥 𝑤 𝑦 𝑧 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 setrec2.1 . . 3 𝑎𝐹
2 nfcv 2905 . . . . . 6 𝑎𝑥
3 nfcv 2905 . . . . . 6 𝑎𝑢
42, 1, 3nfbr 5190 . . . . 5 𝑎 𝑥𝐹𝑢
54nfeuw 2593 . . . 4 𝑎∃!𝑢 𝑥𝐹𝑢
65nfab 2911 . . 3 𝑎{𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}
71, 6nfres 5999 . 2 𝑎(𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})
8 setrec2.2 . . 3 𝐵 = setrecs(𝐹)
9 setrec2lem1 49212 . . . . . . . . . . . 12 ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) = (𝐹𝑤)
109sseq1i 4012 . . . . . . . . . . 11 (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧 ↔ (𝐹𝑤) ⊆ 𝑧)
1110imbi2i 336 . . . . . . . . . 10 ((𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧) ↔ (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
1211imbi2i 336 . . . . . . . . 9 ((𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ (𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)))
1312albii 1819 . . . . . . . 8 (∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)))
1413imbi1i 349 . . . . . . 7 ((∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧))
1514albii 1819 . . . . . 6 (∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧))
1615abbii 2809 . . . . 5 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1716unieqi 4919 . . . 4 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
18 df-setrecs 49203 . . . 4 setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
19 df-setrecs 49203 . . . 4 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
2017, 18, 193eqtr4ri 2776 . . 3 setrecs(𝐹) = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}))
218, 20eqtri 2765 . 2 𝐵 = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}))
22 setrec2lem2 49213 . 2 Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})
23 setrec2.3 . . 3 (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
24 setrec2lem1 49212 . . . . . 6 ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) = (𝐹𝑎)
2524sseq1i 4012 . . . . 5 (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶 ↔ (𝐹𝑎) ⊆ 𝐶)
2625imbi2i 336 . . . 4 ((𝑎𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ (𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
2726albii 1819 . . 3 (∀𝑎(𝑎𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
2823, 27sylibr 234 . 2 (𝜑 → ∀𝑎(𝑎𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶))
297, 21, 22, 28setrec2fun 49211 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  ∃!weu 2568  {cab 2714  wnfc 2890  wss 3951   cuni 4907   class class class wbr 5143  cres 5687  cfv 6561  setrecscsetrecs 49202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-setrecs 49203
This theorem is referenced by:  setrec2v  49215  setrec2mpt  49216
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