Theorem nfun 4117

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 2144, ax-11 2160, ax-12 2180. (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 4100	. . 3 ⊢ (𝑦 ∈ (𝐴 ∪ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵))
2		nfun.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2886	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfun.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2886	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	nfor 1905	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)
7	1, 6	nfxfr 1854	. 2 ⊢ Ⅎ𝑥 𝑦 ∈ (𝐴 ∪ 𝐵)
8	7	nfci 2882	1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 847   ∈ wcel 2111  Ⅎwnfc 2879   ∪ cun 3895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-v 3438  df-un 3902

This theorem is referenced by:  nfsymdif  4204  csbun  4388  iunxdif3  5041  nfsuc  6380  nfsup  9335  nfdju  9800  iunconn  23343  nosupbnd2  27655  noinfbnd2  27670  ordtconnlem1  33937  esumsplit  34066  measvuni  34227  bnj958  34952  bnj1000  34953  bnj1408  35048  bnj1446  35057  bnj1447  35058  bnj1448  35059  bnj1466  35065  bnj1467  35066  rdgssun  37420  exrecfnlem  37421  poimirlem16  37684  poimirlem19  37687  pimxrneun  45534  pimrecltpos  46754