Theorem nfsetrecs 50304

Description: Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.)

Hypothesis

Ref	Expression
nfsetrecs.1	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
nfsetrecs	⊢ Ⅎ𝑥setrecs(𝐹)

Proof of Theorem nfsetrecs

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-setrecs 50302	. 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
2		nfv 1934	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑦
3		nfv 1934	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑧
4		nfsetrecs.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
5		nfcv 2924	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
6	4, 5	nffv 6877	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤)
7		nfcv 2924	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
8	6, 7	nfss 3929	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑤) ⊆ 𝑧
9	3, 8	nfim 1916	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)
10	2, 9	nfim 1916	. . . . . . 7 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
11	10	nfal 2355	. . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
12		nfv 1934	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑧
13	11, 12	nfim 1916	. . . . 5 ⊢ Ⅎ𝑥(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)
14	13	nfal 2355	. . . 4 ⊢ Ⅎ𝑥∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)
15	14	nfab 2930	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
16	15	nfuni 4872	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
17	1, 16	nfcxfr 2922	1 ⊢ Ⅎ𝑥setrecs(𝐹)

Syntax hints:   → wi 4  ∀wal 1558  {cab 2740  Ⅎwnfc 2909   ⊆ wss 3904  ∪ cuni 4865  ‘cfv 6521  setrecscsetrecs 50301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-setrecs 50302

This theorem is referenced by: (None)