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Theorem setrecseq 47891
Description: Equality theorem for set recursion. (Contributed by Emmett Weisz, 17-Feb-2021.)
Assertion
Ref Expression
setrecseq (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))

Proof of Theorem setrecseq
Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq1 6890 . . . . . . . . . 10 (𝐹 = 𝐺 → (𝐹𝑤) = (𝐺𝑤))
21sseq1d 4013 . . . . . . . . 9 (𝐹 = 𝐺 → ((𝐹𝑤) ⊆ 𝑧 ↔ (𝐺𝑤) ⊆ 𝑧))
32imbi2d 340 . . . . . . . 8 (𝐹 = 𝐺 → ((𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧) ↔ (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)))
43imbi2d 340 . . . . . . 7 (𝐹 = 𝐺 → ((𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ (𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧))))
54albidv 1922 . . . . . 6 (𝐹 = 𝐺 → (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧))))
65imbi1d 341 . . . . 5 (𝐹 = 𝐺 → ((∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)))
76albidv 1922 . . . 4 (𝐹 = 𝐺 → (∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)))
87abbidv 2800 . . 3 (𝐹 = 𝐺 → {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
98unieqd 4922 . 2 (𝐹 = 𝐺 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
10 df-setrecs 47890 . 2 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
11 df-setrecs 47890 . 2 setrecs(𝐺) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
129, 10, 113eqtr4g 2796 1 (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538   = wceq 1540  {cab 2708  wss 3948   cuni 4908  cfv 6543  setrecscsetrecs 47889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-setrecs 47890
This theorem is referenced by: (None)
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