Theorem nfsetrecs 50612

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 50610	. 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
2		nfv 1947	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑦
3		nfv 1947	. . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑧
4		nfsetrecs.1	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
5		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤
6	4, 5	nffv 6888	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤)
7		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧
8	6, 7	nfss 3924	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑤) ⊆ 𝑧
9	3, 8	nfim 1929	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)
10	2, 9	nfim 1929	. . . . . . 7 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
11	10	nfal 2353	. . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧))
12		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑧
13	11, 12	nfim 1929	. . . . 5 ⊢ Ⅎ𝑥(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)
14	13	nfal 2353	. . . 4 ⊢ Ⅎ𝑥∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)
15	14	nfab 2928	. . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
16	15	nfuni 4874	. 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)}
17	1, 16	nfcxfr 2920	1 ⊢ Ⅎ𝑥setrecs(𝐹)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  {cab 2738  Ⅎwnfc 2907   ⊆ wss 3899  ∪ cuni 4867  ‘cfv 6533  setrecscsetrecs 50609

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-setrecs 50610

This theorem is used by: (None)