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

Theorem nfixpw 8481
 Description: Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8482 with a disjoint variable condition, which does not require ax-13 2379. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Gino Giotto, 26-Jan-2024.)
Hypotheses
Ref Expression
nfixpw.1 𝑦𝐴
nfixpw.2 𝑦𝐵
Assertion
Ref Expression
nfixpw 𝑦X𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfixpw
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ixp 8463 . 2 X𝑥𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
2 nfcv 2955 . . . . 5 𝑦𝑧
3 nfcv 2955 . . . . . . . . 9 𝑦𝑥
4 nfixpw.1 . . . . . . . . 9 𝑦𝐴
53, 4nfel 2969 . . . . . . . 8 𝑦 𝑥𝐴
65nfab 2961 . . . . . . 7 𝑦{𝑥𝑥𝐴}
76a1i 11 . . . . . 6 (⊤ → 𝑦{𝑥𝑥𝐴})
87mptru 1545 . . . . 5 𝑦{𝑥𝑥𝐴}
92, 8nffn 6430 . . . 4 𝑦 𝑧 Fn {𝑥𝑥𝐴}
10 df-ral 3111 . . . . 5 (∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
11 nftru 1806 . . . . . . 7 𝑥
125a1i 11 . . . . . . . 8 (⊤ → Ⅎ𝑦 𝑥𝐴)
132a1i 11 . . . . . . . . . 10 (⊤ → 𝑦𝑧)
143a1i 11 . . . . . . . . . 10 (⊤ → 𝑦𝑥)
1513, 14nffvd 6667 . . . . . . . . 9 (⊤ → 𝑦(𝑧𝑥))
16 nfixpw.2 . . . . . . . . . 10 𝑦𝐵
1716a1i 11 . . . . . . . . 9 (⊤ → 𝑦𝐵)
1815, 17nfeld 2966 . . . . . . . 8 (⊤ → Ⅎ𝑦(𝑧𝑥) ∈ 𝐵)
1912, 18nfimd 1895 . . . . . . 7 (⊤ → Ⅎ𝑦(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2011, 19nfald 2336 . . . . . 6 (⊤ → Ⅎ𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2120mptru 1545 . . . . 5 𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵)
2210, 21nfxfr 1854 . . . 4 𝑦𝑥𝐴 (𝑧𝑥) ∈ 𝐵
239, 22nfan 1900 . . 3 𝑦(𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)
2423nfab 2961 . 2 𝑦{𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
251, 24nfcxfr 2953 1 𝑦X𝑥𝐴 𝐵
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ⊤wtru 1539  Ⅎwnf 1785   ∈ wcel 2111  {cab 2776  Ⅎwnfc 2936  ∀wral 3106   Fn wfn 6327  ‘cfv 6332  Xcixp 8462 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6291  df-fun 6334  df-fn 6335  df-fv 6340  df-ixp 8463 This theorem is referenced by:  vonioo  43489
 Copyright terms: Public domain W3C validator