Theorem nfsetrecs 48917

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 48915	. 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
2		nfv 1912	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑦
3		nfv 1912	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑧
4		nfsetrecs.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
5		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
6	4, 5	nffv 6917	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤)
7		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
8	6, 7	nfss 3988	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑤) ⊆ 𝑧
9	3, 8	nfim 1894	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)
10	2, 9	nfim 1894	. . . . . . 7 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
11	10	nfal 2322	. . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
12		nfv 1912	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑧
13	11, 12	nfim 1894	. . . . 5 ⊢ Ⅎ𝑥(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)
14	13	nfal 2322	. . . 4 ⊢ Ⅎ𝑥∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)
15	14	nfab 2909	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
16	15	nfuni 4919	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
17	1, 16	nfcxfr 2901	1 ⊢ Ⅎ𝑥setrecs(𝐹)

Syntax hints:   → wi 4  ∀wal 1535  {cab 2712  Ⅎwnfc 2888   ⊆ wss 3963  ∪ cuni 4912  ‘cfv 6563  setrecscsetrecs 48914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-setrecs 48915

This theorem is referenced by: (None)