Theorem nfsetrecs 47721

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 47719	. 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
2		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑦
3		nfv 1917	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑧
4		nfsetrecs.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
5		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
6	4, 5	nffv 6901	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤)
7		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
8	6, 7	nfss 3974	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑤) ⊆ 𝑧
9	3, 8	nfim 1899	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)
10	2, 9	nfim 1899	. . . . . . 7 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
11	10	nfal 2316	. . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
12		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑧
13	11, 12	nfim 1899	. . . . 5 ⊢ Ⅎ𝑥(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)
14	13	nfal 2316	. . . 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 4915	. 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 1539  {cab 2709  Ⅎwnfc 2883   ⊆ wss 3948  ∪ cuni 4908  ‘cfv 6543  setrecscsetrecs 47718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-setrecs 47719

This theorem is referenced by: (None)