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Theorem setrecseq 45256
 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 6645 . . . . . . . . . 10 (𝐹 = 𝐺 → (𝐹𝑤) = (𝐺𝑤))
21sseq1d 3946 . . . . . . . . 9 (𝐹 = 𝐺 → ((𝐹𝑤) ⊆ 𝑧 ↔ (𝐺𝑤) ⊆ 𝑧))
32imbi2d 344 . . . . . . . 8 (𝐹 = 𝐺 → ((𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧) ↔ (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)))
43imbi2d 344 . . . . . . 7 (𝐹 = 𝐺 → ((𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ (𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧))))
54albidv 1921 . . . . . 6 (𝐹 = 𝐺 → (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧))))
65imbi1d 345 . . . . 5 (𝐹 = 𝐺 → ((∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)))
76albidv 1921 . . . 4 (𝐹 = 𝐺 → (∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)))
87abbidv 2862 . . 3 (𝐹 = 𝐺 → {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
98unieqd 4815 . 2 (𝐹 = 𝐺 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
10 df-setrecs 45255 . 2 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
11 df-setrecs 45255 . 2 setrecs(𝐺) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
129, 10, 113eqtr4g 2858 1 (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   = wceq 1538  {cab 2776   ⊆ wss 3881  ∪ cuni 4801  ‘cfv 6325  setrecscsetrecs 45254 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-uni 4802  df-br 5032  df-iota 6284  df-fv 6333  df-setrecs 45255 This theorem is referenced by: (None)
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