Theorem nfun 4133

Description: Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) Avoid ax-10 2142, ax-11 2158, ax-12 2178. (Revised by SN, 14-May-2025.)

Hypotheses

Ref	Expression
nfun.1	⊢ Ⅎ𝑥𝐴
nfun.2	⊢ Ⅎ𝑥𝐵

Assertion

Ref	Expression
nfun	⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Proof of Theorem nfun

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 4116	. . 3 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵))
2		nfun.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2883	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfun.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2883	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	nfor 1904	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)
7	1, 6	nfxfr 1853	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∪ 𝐵)
8	7	nfci 2879	1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 847   ∈ wcel 2109  Ⅎwnfc 2876   ∪ cun 3912

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-v 3449  df-un 3919

This theorem is referenced by:  nfsymdif  4220  csbun  4404  iunxdif3  5059  nfsuc  6406  nfsup  9402  nfdju  9860  iunconn  23315  nosupbnd2  27628  noinfbnd2  27643  ordtconnlem1  33914  esumsplit  34043  measvuni  34204  bnj958  34930  bnj1000  34931  bnj1408  35026  bnj1446  35035  bnj1447  35036  bnj1448  35037  bnj1466  35043  bnj1467  35044  rdgssun  37366  exrecfnlem  37367  poimirlem16  37630  poimirlem19  37633  pimxrneun  45484  pimrecltpos  46706