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Theorem setrec2 47740
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 47736 and setrec2v 47741 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 7842) 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 2904 . . . . . 6 𝑎𝑥
3 nfcv 2904 . . . . . 6 𝑎𝑢
42, 1, 3nfbr 5196 . . . . 5 𝑎 𝑥𝐹𝑢
54nfeuw 2588 . . . 4 𝑎∃!𝑢 𝑥𝐹𝑢
65nfab 2910 . . 3 𝑎{𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}
71, 6nfres 5984 . 2 𝑎(𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})
8 setrec2.2 . . 3 𝐵 = setrecs(𝐹)
9 setrec2lem1 47738 . . . . . . . . . . . 12 ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) = (𝐹𝑤)
109sseq1i 4011 . . . . . . . . . . 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 1822 . . . . . . . 8 (∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)))
1413imbi1i 350 . . . . . . 7 ((∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧))
1514albii 1822 . . . . . 6 (∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧))
1615abbii 2803 . . . . 5 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1716unieqi 4922 . . . 4 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
18 df-setrecs 47729 . . . 4 setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
19 df-setrecs 47729 . . . 4 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
2017, 18, 193eqtr4ri 2772 . . 3 setrecs(𝐹) = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}))
218, 20eqtri 2761 . 2 𝐵 = setrecs((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢}))
22 setrec2lem2 47739 . 2 Fun (𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})
23 setrec2.3 . . 3 (𝜑 → ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
24 setrec2lem1 47738 . . . . . 6 ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) = (𝐹𝑎)
2524sseq1i 4011 . . . . 5 (((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶 ↔ (𝐹𝑎) ⊆ 𝐶)
2625imbi2i 336 . . . 4 ((𝑎𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ (𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
2726albii 1822 . . 3 (∀𝑎(𝑎𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶) ↔ ∀𝑎(𝑎𝐶 → (𝐹𝑎) ⊆ 𝐶))
2823, 27sylibr 233 . 2 (𝜑 → ∀𝑎(𝑎𝐶 → ((𝐹 ↾ {𝑥 ∣ ∃!𝑢 𝑥𝐹𝑢})‘𝑎) ⊆ 𝐶))
297, 21, 22, 28setrec2fun 47737 1 (𝜑𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540   = wceq 1542  ∃!weu 2563  {cab 2710  wnfc 2884  wss 3949   cuni 4909   class class class wbr 5149  cres 5679  cfv 6544  setrecscsetrecs 47728
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-setrecs 47729
This theorem is referenced by:  setrec2v  47741  setrec2mpt  47742
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