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Theorem nfsetrecs 43001
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 42999 . 2 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
2 nfv 2005 . . . . . . . 8 𝑥 𝑤𝑦
3 nfv 2005 . . . . . . . . 9 𝑥 𝑤𝑧
4 nfsetrecs.1 . . . . . . . . . . 11 𝑥𝐹
5 nfcv 2948 . . . . . . . . . . 11 𝑥𝑤
64, 5nffv 6418 . . . . . . . . . 10 𝑥(𝐹𝑤)
7 nfcv 2948 . . . . . . . . . 10 𝑥𝑧
86, 7nfss 3791 . . . . . . . . 9 𝑥(𝐹𝑤) ⊆ 𝑧
93, 8nfim 1987 . . . . . . . 8 𝑥(𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)
102, 9nfim 1987 . . . . . . 7 𝑥(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
1110nfal 2329 . . . . . 6 𝑥𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
12 nfv 2005 . . . . . 6 𝑥 𝑦𝑧
1311, 12nfim 1987 . . . . 5 𝑥(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)
1413nfal 2329 . . . 4 𝑥𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)
1514nfab 2953 . . 3 𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1615nfuni 4636 . 2 𝑥 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
171, 16nfcxfr 2946 1 𝑥setrecs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1635  {cab 2792  wnfc 2935  wss 3769   cuni 4630  cfv 6101  setrecscsetrecs 42998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-iota 6064  df-fv 6109  df-setrecs 42999
This theorem is referenced by: (None)
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