Theorem nfiun 4580

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.)

Hypotheses

Ref	Expression
nfiun.1	⊢ Ⅎ𝑦𝐴
nfiun.2	⊢ Ⅎ𝑦𝐵

Assertion

Ref	Expression
nfiun	⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Proof of Theorem nfiun

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4554	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiun.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiun.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2787	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfrex 3036	. . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab 2798	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2791	1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∈ wcel 2030  {cab 2637  Ⅎwnfc 2780  ∃wrex 2942  ∪ ciun 4552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-iun 4554

This theorem is referenced by:  iunab  4598  disjxiun  4681  disjxiunOLD  4682  ovoliunnul  23321  iundisjf  29528  iundisj2f  29529  iundisjfi  29683  iundisj2fi  29684  bnj1498  31255  trpredlem1  31851  trpredrec  31862  ss2iundf  38268  fnlimcnv  40217  fnlimfvre  40224  fnlimabslt  40229  smfaddlem1  41292  smflimlem6  41305  smflim  41306  smfmullem4  41322  smflim2  41333  smflimsup  41355  smfliminf  41358