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Theorem nfixpw 8955
Description: Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8956 with a disjoint variable condition, which does not require ax-13 2375. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2375. (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 8937 . 2 X𝑥𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
2 nfcv 2903 . . . . 5 𝑦𝑧
3 nfcv 2903 . . . . . . . . 9 𝑦𝑥
4 nfixpw.1 . . . . . . . . 9 𝑦𝐴
53, 4nfel 2918 . . . . . . . 8 𝑦 𝑥𝐴
65nfab 2909 . . . . . . 7 𝑦{𝑥𝑥𝐴}
76a1i 11 . . . . . 6 (⊤ → 𝑦{𝑥𝑥𝐴})
87mptru 1544 . . . . 5 𝑦{𝑥𝑥𝐴}
92, 8nffn 6668 . . . 4 𝑦 𝑧 Fn {𝑥𝑥𝐴}
10 df-ral 3060 . . . . 5 (∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
11 nftru 1801 . . . . . . 7 𝑥
125a1i 11 . . . . . . . 8 (⊤ → Ⅎ𝑦 𝑥𝐴)
132a1i 11 . . . . . . . . . 10 (⊤ → 𝑦𝑧)
143a1i 11 . . . . . . . . . 10 (⊤ → 𝑦𝑥)
1513, 14nffvd 6919 . . . . . . . . 9 (⊤ → 𝑦(𝑧𝑥))
16 nfixpw.2 . . . . . . . . . 10 𝑦𝐵
1716a1i 11 . . . . . . . . 9 (⊤ → 𝑦𝐵)
1815, 17nfeld 2915 . . . . . . . 8 (⊤ → Ⅎ𝑦(𝑧𝑥) ∈ 𝐵)
1912, 18nfimd 1892 . . . . . . 7 (⊤ → Ⅎ𝑦(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2011, 19nfald 2327 . . . . . 6 (⊤ → Ⅎ𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2120mptru 1544 . . . . 5 𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵)
2210, 21nfxfr 1850 . . . 4 𝑦𝑥𝐴 (𝑧𝑥) ∈ 𝐵
239, 22nfan 1897 . . 3 𝑦(𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)
2423nfab 2909 . 2 𝑦{𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
251, 24nfcxfr 2901 1 𝑦X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535  wtru 1538  wnf 1780  wcel 2106  {cab 2712  wnfc 2888  wral 3059   Fn wfn 6558  cfv 6563  Xcixp 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571  df-ixp 8937
This theorem is referenced by:  vonioo  46638
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