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 2402. 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 2915	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfrexw 3311	. . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab 2929	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2921	1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

This proof depends on syntax axioms:   ∈ wcel 2145  {cab 2739  Ⅎwnfc 2908  ∃wrex 3087  ∪ 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 2733

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 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-iun 4953

This theorem is used by:  iunab  5010  disjxiun  5100  ttrclselem1  9726  ttrclselem2  9727  ovoliunnul  25828  iunxpssiun1  33162  iundisjf  33183  iundisj2f  33184  iundisjfi  33388  iundisj2fi  33389  suppgsumssiun  33633  bnj1498  35691  nfttc  37279  ss2iundf  44658  nfcoll  45239  fnlimcnv  46676  fnlimfvre  46683  fnlimabslt  46688  smfaddlem1  47772  smflimlem6  47785  smflim  47786  smfmullem4  47803  smflim2  47815  smflimsup  47837  smfliminf  47840  fsupdm  47851  finfdm  47855