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Theorem nfixpw 8910
Description: Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8911 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2372. (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 8892 . 2 X𝑥𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
2 nfcv 2904 . . . . 5 𝑦𝑧
3 nfcv 2904 . . . . . . . . 9 𝑦𝑥
4 nfixpw.1 . . . . . . . . 9 𝑦𝐴
53, 4nfel 2918 . . . . . . . 8 𝑦 𝑥𝐴
65nfab 2910 . . . . . . 7 𝑦{𝑥𝑥𝐴}
76a1i 11 . . . . . 6 (⊤ → 𝑦{𝑥𝑥𝐴})
87mptru 1549 . . . . 5 𝑦{𝑥𝑥𝐴}
92, 8nffn 6649 . . . 4 𝑦 𝑧 Fn {𝑥𝑥𝐴}
10 df-ral 3063 . . . . 5 (∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
11 nftru 1807 . . . . . . 7 𝑥
125a1i 11 . . . . . . . 8 (⊤ → Ⅎ𝑦 𝑥𝐴)
132a1i 11 . . . . . . . . . 10 (⊤ → 𝑦𝑧)
143a1i 11 . . . . . . . . . 10 (⊤ → 𝑦𝑥)
1513, 14nffvd 6904 . . . . . . . . 9 (⊤ → 𝑦(𝑧𝑥))
16 nfixpw.2 . . . . . . . . . 10 𝑦𝐵
1716a1i 11 . . . . . . . . 9 (⊤ → 𝑦𝐵)
1815, 17nfeld 2915 . . . . . . . 8 (⊤ → Ⅎ𝑦(𝑧𝑥) ∈ 𝐵)
1912, 18nfimd 1898 . . . . . . 7 (⊤ → Ⅎ𝑦(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2011, 19nfald 2322 . . . . . 6 (⊤ → Ⅎ𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2120mptru 1549 . . . . 5 𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵)
2210, 21nfxfr 1856 . . . 4 𝑦𝑥𝐴 (𝑧𝑥) ∈ 𝐵
239, 22nfan 1903 . . 3 𝑦(𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)
2423nfab 2910 . 2 𝑦{𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
251, 24nfcxfr 2902 1 𝑦X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1540  wtru 1543  wnf 1786  wcel 2107  {cab 2710  wnfc 2884  wral 3062   Fn wfn 6539  cfv 6544  Xcixp 8891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-ixp 8892
This theorem is referenced by:  vonioo  45398
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