MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfiin Structured version   Visualization version   GIF version

Theorem nfiin 5025
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2366. See nfiing 5027 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.)
Hypotheses
Ref Expression
nfiun.1 𝑦𝐴
nfiun.2 𝑦𝐵
Assertion
Ref Expression
nfiin 𝑦 𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfiin
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-iin 4997 . 2 𝑥𝐴 𝐵 = {𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
2 nfiun.1 . . . 4 𝑦𝐴
3 nfiun.2 . . . . 5 𝑦𝐵
43nfcri 2883 . . . 4 𝑦 𝑧𝐵
52, 4nfralw 3299 . . 3 𝑦𝑥𝐴 𝑧𝐵
65nfab 2898 . 2 𝑦{𝑧 ∣ ∀𝑥𝐴 𝑧𝐵}
71, 6nfcxfr 2890 1 𝑦 𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wcel 2099  {cab 2703  wnfc 2876  wral 3051   ciin 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-iin 4997
This theorem is referenced by:  iinab  5069  fnlimcnv  45322  fnlimfvre  45329  fnlimabslt  45334  iinhoiicc  46329  preimageiingt  46375  preimaleiinlt  46376  smflimlem6  46431  smflim  46432  smflim2  46461  smfsup  46469  smfsupmpt  46470  smfsupxr  46471  smfinflem  46472  smfinf  46473  smfinfmpt  46474  smflimsup  46483  smfliminf  46486  fsupdm  46497  finfdm  46501
  Copyright terms: Public domain W3C validator