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

Theorem nfixpw 8956
Description: Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8957 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2377. (Revised by GG, 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 8938 . 2 X𝑥𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
2 nfcv 2905 . . . . 5 𝑦𝑧
3 nfcv 2905 . . . . . . . . 9 𝑦𝑥
4 nfixpw.1 . . . . . . . . 9 𝑦𝐴
53, 4nfel 2920 . . . . . . . 8 𝑦 𝑥𝐴
65nfab 2911 . . . . . . 7 𝑦{𝑥𝑥𝐴}
76a1i 11 . . . . . 6 (⊤ → 𝑦{𝑥𝑥𝐴})
87mptru 1547 . . . . 5 𝑦{𝑥𝑥𝐴}
92, 8nffn 6667 . . . 4 𝑦 𝑧 Fn {𝑥𝑥𝐴}
10 df-ral 3062 . . . . 5 (∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
11 nftru 1804 . . . . . . 7 𝑥
125a1i 11 . . . . . . . 8 (⊤ → Ⅎ𝑦 𝑥𝐴)
132a1i 11 . . . . . . . . . 10 (⊤ → 𝑦𝑧)
143a1i 11 . . . . . . . . . 10 (⊤ → 𝑦𝑥)
1513, 14nffvd 6918 . . . . . . . . 9 (⊤ → 𝑦(𝑧𝑥))
16 nfixpw.2 . . . . . . . . . 10 𝑦𝐵
1716a1i 11 . . . . . . . . 9 (⊤ → 𝑦𝐵)
1815, 17nfeld 2917 . . . . . . . 8 (⊤ → Ⅎ𝑦(𝑧𝑥) ∈ 𝐵)
1912, 18nfimd 1894 . . . . . . 7 (⊤ → Ⅎ𝑦(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2011, 19nfald 2328 . . . . . 6 (⊤ → Ⅎ𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2120mptru 1547 . . . . 5 𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵)
2210, 21nfxfr 1853 . . . 4 𝑦𝑥𝐴 (𝑧𝑥) ∈ 𝐵
239, 22nfan 1899 . . 3 𝑦(𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)
2423nfab 2911 . 2 𝑦{𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
251, 24nfcxfr 2903 1 𝑦X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538  wtru 1541  wnf 1783  wcel 2108  {cab 2714  wnfc 2890  wral 3061   Fn wfn 6556  cfv 6561  Xcixp 8937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ixp 8938
This theorem is referenced by:  vonioo  46697
  Copyright terms: Public domain W3C validator