Theorem nfsetrecs 49205

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 49203	. 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
2		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑦
3		nfv 1914	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑧
4		nfsetrecs.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
5		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
6	4, 5	nffv 6916	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤)
7		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
8	6, 7	nfss 3976	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑤) ⊆ 𝑧
9	3, 8	nfim 1896	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)
10	2, 9	nfim 1896	. . . . . . 7 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
11	10	nfal 2323	. . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
12		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑧
13	11, 12	nfim 1896	. . . . 5 ⊢ Ⅎ𝑥(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)
14	13	nfal 2323	. . . 4 ⊢ Ⅎ𝑥∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)
15	14	nfab 2911	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
16	15	nfuni 4914	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
17	1, 16	nfcxfr 2903	1 ⊢ Ⅎ𝑥setrecs(𝐹)

Syntax hints:   → wi 4  ∀wal 1538  {cab 2714  Ⅎwnfc 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)