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Theorem nfsetrecs 44291
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 44289 . 2 setrecs(𝐹) = {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
2 nfv 1896 . . . . . . . 8 𝑥 𝑤𝑦
3 nfv 1896 . . . . . . . . 9 𝑥 𝑤𝑧
4 nfsetrecs.1 . . . . . . . . . . 11 𝑥𝐹
5 nfcv 2951 . . . . . . . . . . 11 𝑥𝑤
64, 5nffv 6555 . . . . . . . . . 10 𝑥(𝐹𝑤)
7 nfcv 2951 . . . . . . . . . 10 𝑥𝑧
86, 7nfss 3888 . . . . . . . . 9 𝑥(𝐹𝑤) ⊆ 𝑧
93, 8nfim 1882 . . . . . . . 8 𝑥(𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)
102, 9nfim 1882 . . . . . . 7 𝑥(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
1110nfal 2307 . . . . . 6 𝑥𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧))
12 nfv 1896 . . . . . 6 𝑥 𝑦𝑧
1311, 12nfim 1882 . . . . 5 𝑥(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)
1413nfal 2307 . . . 4 𝑥𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)
1514nfab 2957 . . 3 𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
1615nfuni 4757 . 2 𝑥 {𝑦 ∣ ∀𝑧(∀𝑤(𝑤𝑦 → (𝑤𝑧 → (𝐹𝑤) ⊆ 𝑧)) → 𝑦𝑧)}
171, 16nfcxfr 2949 1 𝑥setrecs(𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1523  {cab 2777  wnfc 2935  wss 3865   cuni 4751  cfv 6232  setrecscsetrecs 44288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-iota 6196  df-fv 6240  df-setrecs 44289
This theorem is referenced by: (None)
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