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Theorem nfsetrecs 47217
Description: Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)
Hypothesis
Ref Expression
nfsetrecs.1 𝑥𝐹
Assertion
Ref Expression
nfsetrecs 𝑥setrecs(𝐹)

Proof of Theorem nfsetrecs
Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-setrecs 47215 . 2 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
2 nfv 1918 . . . . . . . 8 𝑥 𝑤𝑦
3 nfv 1918 . . . . . . . . 9 𝑥 𝑤𝑧
4 nfsetrecs.1 . . . . . . . . . . 11 𝑥𝐹
5 nfcv 2904 . . . . . . . . . . 11 𝑥𝑤
64, 5nffv 6853 . . . . . . . . . 10 𝑥(𝐹𝑤)
7 nfcv 2904 . . . . . . . . . 10 𝑥𝑧
86, 7nfss 3937 . . . . . . . . 9 𝑥(𝐹𝑤) ⊆ 𝑧
93, 8nfim 1900 . . . . . . . 8 𝑥(𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)
102, 9nfim 1900 . . . . . . 7 𝑥(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
1110nfal 2317 . . . . . 6 𝑥𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
12 nfv 1918 . . . . . 6 𝑥 𝑦𝑧
1311, 12nfim 1900 . . . . 5 𝑥(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)
1413nfal 2317 . . . 4 𝑥𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)
1514nfab 2910 . . 3 𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1615nfuni 4873 . 2 𝑥 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
171, 16nfcxfr 2902 1 𝑥setrecs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1540  {cab 2710  wnfc 2884  wss 3911   cuni 4866  cfv 6497  setrecscsetrecs 47214
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
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-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-iota 6449  df-fv 6505  df-setrecs 47215
This theorem is referenced by: (None)
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