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Theorem nfsetrecs 49205
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 49203 . 2 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
2 nfv 1914 . . . . . . . 8 𝑥 𝑤𝑦
3 nfv 1914 . . . . . . . . 9 𝑥 𝑤𝑧
4 nfsetrecs.1 . . . . . . . . . . 11 𝑥𝐹
5 nfcv 2905 . . . . . . . . . . 11 𝑥𝑤
64, 5nffv 6916 . . . . . . . . . 10 𝑥(𝐹𝑤)
7 nfcv 2905 . . . . . . . . . 10 𝑥𝑧
86, 7nfss 3976 . . . . . . . . 9 𝑥(𝐹𝑤) ⊆ 𝑧
93, 8nfim 1896 . . . . . . . 8 𝑥(𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)
102, 9nfim 1896 . . . . . . 7 𝑥(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
1110nfal 2323 . . . . . 6 𝑥𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
12 nfv 1914 . . . . . 6 𝑥 𝑦𝑧
1311, 12nfim 1896 . . . . 5 𝑥(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)
1413nfal 2323 . . . 4 𝑥𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)
1514nfab 2911 . . 3 𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1615nfuni 4914 . 2 𝑥 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
171, 16nfcxfr 2903 1 𝑥setrecs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1538  {cab 2714  wnfc 2890  wss 3951   cuni 4907  cfv 6561  setrecscsetrecs 49202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-setrecs 49203
This theorem is referenced by: (None)
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