Theorem nfun 3769

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 3579	. 2 ⊢ (𝐴 ∪ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
2		nfun.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
3	2	nfcri 2758	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfun.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
5	4	nfcri 2758	. . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵
6	3, 5	nfor 1834	. . 3 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)
7	6	nfab 2769	. 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵)}
8	1, 7	nfcxfr 2762	1 ⊢ Ⅎ𝑥(𝐴 ∪ 𝐵)

Syntax hints:   ∨ wo 383   ∈ wcel 1990  {cab 2608  Ⅎwnfc 2751   ∪ cun 3572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-un 3579

This theorem is referenced by:  nfsymdif  3848  csbun  4009  iunxdif3  4606  nfsuc  5796  nfsup  8357  iunconn  21231  ordtconnlem1  29970  esumsplit  30115  measvuni  30277  bnj958  31010  bnj1000  31011  bnj1408  31104  bnj1446  31113  bnj1447  31114  bnj1448  31115  bnj1466  31121  bnj1467  31122  nosupbnd2  31862  poimirlem16  33425  poimirlem19  33428  pimrecltpos  40919