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Theorem setrecseq 44716
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 6662 . . . . . . . . . 10 (𝐹 = 𝐺 → (𝐹𝑤) = (𝐺𝑤))
21sseq1d 3995 . . . . . . . . 9 (𝐹 = 𝐺 → ((𝐹𝑤) ⊆ 𝑧 ↔ (𝐺𝑤) ⊆ 𝑧))
32imbi2d 342 . . . . . . . 8 (𝐹 = 𝐺 → ((𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧) ↔ (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)))
43imbi2d 342 . . . . . . 7 (𝐹 = 𝐺 → ((𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ (𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧))))
54albidv 1912 . . . . . 6 (𝐹 = 𝐺 → (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧))))
65imbi1d 343 . . . . 5 (𝐹 = 𝐺 → ((∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)))
76albidv 1912 . . . 4 (𝐹 = 𝐺 → (∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)))
87abbidv 2882 . . 3 (𝐹 = 𝐺 → {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
98unieqd 4840 . 2 (𝐹 = 𝐺 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
10 df-setrecs 44715 . 2 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
11 df-setrecs 44715 . 2 setrecs(𝐺) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
129, 10, 113eqtr4g 2878 1 (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1526   = wceq 1528  {cab 2796  wss 3933   cuni 4830  cfv 6348  setrecscsetrecs 44714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-rex 3141  df-in 3940  df-ss 3949  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-setrecs 44715
This theorem is referenced by: (None)
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