Theorem nfiun 4982

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2401. See nfiung 4984 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.)

Hypotheses

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

Assertion

Ref	Expression
nfiun	⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥, 𝑦)   𝐵(𝑥, 𝑦)

Proof of Theorem nfiun

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4953	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiun.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiun.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2914	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfrexw 3310	. . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab 2928	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2920	1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

This proof depends on syntax axioms:   ∈ wcel 2145  {cab 2738  Ⅎwnfc 2907  ∃wrex 3086  ∪ ciun 4951

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-iun 4953

This theorem is used by:  iunab  5010  disjxiun  5100  ttrclselem1  9707  ttrclselem2  9708  ovoliunnul  25738  iunxpssiun1  33044  iundisjf  33065  iundisj2f  33066  iundisjfi  33270  iundisj2fi  33271  suppgsumssiun  33515  bnj1498  35573  nfttc  37113  ss2iundf  44502  nfcoll  45083  fnlimcnv  46498  fnlimfvre  46505  fnlimabslt  46510  smfaddlem1  47594  smflimlem6  47607  smflim  47608  smfmullem4  47625  smflim2  47637  smflimsup  47659  smfliminf  47662  fsupdm  47673  finfdm  47677