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Theorem setrecseq 42950
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 6331 . . . . . . . . . 10 (𝐹 = 𝐺 → (𝐹𝑤) = (𝐺𝑤))
21sseq1d 3779 . . . . . . . . 9 (𝐹 = 𝐺 → ((𝐹𝑤) ⊆ 𝑧 ↔ (𝐺𝑤) ⊆ 𝑧))
32imbi2d 329 . . . . . . . 8 (𝐹 = 𝐺 → ((𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧) ↔ (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)))
43imbi2d 329 . . . . . . 7 (𝐹 = 𝐺 → ((𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ (𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧))))
54albidv 2000 . . . . . 6 (𝐹 = 𝐺 → (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) ↔ ∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧))))
65imbi1d 330 . . . . 5 (𝐹 = 𝐺 → ((∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ (∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)))
76albidv 2000 . . . 4 (𝐹 = 𝐺 → (∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧) ↔ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)))
87abbidv 2889 . . 3 (𝐹 = 𝐺 → {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
98unieqd 4582 . 2 (𝐹 = 𝐺 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)} = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)})
10 df-setrecs 42949 . 2 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
11 df-setrecs 42949 . 2 setrecs(𝐺) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐺𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
129, 10, 113eqtr4g 2829 1 (𝐹 = 𝐺 → setrecs(𝐹) = setrecs(𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1628   = wceq 1630  {cab 2756  wss 3721   cuni 4572  cfv 6031  setrecscsetrecs 42948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-in 3728  df-ss 3735  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-setrecs 42949
This theorem is referenced by: (None)
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