Theorem nfiun 4985

Description: Bound-variable hypothesis builder for indexed union. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2404. See nfiung 4987 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 4956	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		nfiun.1	. . . 4 ⊢ Ⅎ𝑦𝐴
3		nfiun.2	. . . . 5 ⊢ Ⅎ𝑦𝐵
4	3	nfcri 2918	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵
5	2, 4	nfrexw 3314	. . 3 ⊢ Ⅎ𝑦∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵
6	5	nfab 2932	. 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
7	1, 6	nfcxfr 2924	1 ⊢ Ⅎ𝑦∪ 𝑥 ∈ 𝐴 𝐵

This proof depends on syntax axioms:   ∈ wcel 2146  {cab 2741  Ⅎwnfc 2911  ∃wrex 3090  ∪ ciun 4954

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-iun 4956

This theorem is used by:  iunab  5013  disjxiun  5103  ttrclselem1  9739  ttrclselem2  9740  ovoliunnul  25843  iunxpssiun1  33219  iundisjf  33240  iundisj2f  33241  iundisjfi  33444  iundisj2fi  33445  suppgsumssiun  33689  bnj1498  35747  nfttc  37409  ss2iundf  44802  nfcoll  45383  fnlimcnv  46819  fnlimfvre  46826  fnlimabslt  46831  smfaddlem1  47915  smflimlem6  47928  smflim  47929  smfmullem4  47946  smflim2  47958  smflimsup  47980  smfliminf  47983  fsupdm  47994  finfdm  47998