Theorem nfun 4166

Description: Bound-variable hypothesis builder for the union of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.)

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		df-un 3954	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
2		nfun.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2891	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfun.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2891	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	nfor 1908	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)
7	6	nfab 2910	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
8	1, 7	nfcxfr 2902	1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 846   ∈ wcel 2107  {cab 2710  Ⅎwnfc 2884   ∪ cun 3947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-un 3954

This theorem is referenced by:  nfsymdif  4247  csbun  4439  iunxdif3  5099  nfsuc  6437  nfsup  9446  nfdju  9902  iunconn  22932  nosupbnd2  27219  noinfbnd2  27234  ordtconnlem1  32904  esumsplit  33051  measvuni  33212  bnj958  33951  bnj1000  33952  bnj1408  34047  bnj1446  34056  bnj1447  34057  bnj1448  34058  bnj1466  34064  bnj1467  34065  rdgssun  36259  exrecfnlem  36260  poimirlem16  36504  poimirlem19  36507  pimxrneun  44199  pimrecltpos  45424