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Theorem nfsetrecs 42951
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 42949 . 2 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
2 nfv 1994 . . . . . . . 8 𝑥 𝑤𝑦
3 nfv 1994 . . . . . . . . 9 𝑥 𝑤𝑧
4 nfsetrecs.1 . . . . . . . . . . 11 𝑥𝐹
5 nfcv 2912 . . . . . . . . . . 11 𝑥𝑤
64, 5nffv 6339 . . . . . . . . . 10 𝑥(𝐹𝑤)
7 nfcv 2912 . . . . . . . . . 10 𝑥𝑧
86, 7nfss 3743 . . . . . . . . 9 𝑥(𝐹𝑤) ⊆ 𝑧
93, 8nfim 1976 . . . . . . . 8 𝑥(𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)
102, 9nfim 1976 . . . . . . 7 𝑥(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
1110nfal 2316 . . . . . 6 𝑥𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
12 nfv 1994 . . . . . 6 𝑥 𝑦𝑧
1311, 12nfim 1976 . . . . 5 𝑥(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)
1413nfal 2316 . . . 4 𝑥𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)
1514nfab 2917 . . 3 𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1615nfuni 4578 . 2 𝑥 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
171, 16nfcxfr 2910 1 𝑥setrecs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1628  {cab 2756  wnfc 2899  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-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  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-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  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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