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Theorem nfixp 8958
Description: Bound-variable hypothesis builder for indexed Cartesian product. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker nfixpw 8957 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfixp.1 𝑦𝐴
nfixp.2 𝑦𝐵
Assertion
Ref Expression
nfixp 𝑦X𝑥𝐴 𝐵

Proof of Theorem nfixp
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ixp 8939 . 2 X𝑥𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
2 nfcv 2904 . . . . 5 𝑦𝑧
3 nftru 1803 . . . . . . 7 𝑥
4 nfcvf 2931 . . . . . . . . 9 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
54adantl 481 . . . . . . . 8 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝑥)
6 nfixp.1 . . . . . . . . 9 𝑦𝐴
76a1i 11 . . . . . . . 8 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝐴)
85, 7nfeld 2916 . . . . . . 7 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦 𝑥𝐴)
93, 8nfabd2 2928 . . . . . 6 (⊤ → 𝑦{𝑥𝑥𝐴})
109mptru 1546 . . . . 5 𝑦{𝑥𝑥𝐴}
112, 10nffn 6666 . . . 4 𝑦 𝑧 Fn {𝑥𝑥𝐴}
12 df-ral 3061 . . . . 5 (∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
132a1i 11 . . . . . . . . . 10 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝑧)
1413, 5nffvd 6917 . . . . . . . . 9 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦(𝑧𝑥))
15 nfixp.2 . . . . . . . . . 10 𝑦𝐵
1615a1i 11 . . . . . . . . 9 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝐵)
1714, 16nfeld 2916 . . . . . . . 8 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑧𝑥) ∈ 𝐵)
188, 17nfimd 1893 . . . . . . 7 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
193, 18nfald2 2449 . . . . . 6 (⊤ → Ⅎ𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2019mptru 1546 . . . . 5 𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵)
2112, 20nfxfr 1852 . . . 4 𝑦𝑥𝐴 (𝑧𝑥) ∈ 𝐵
2211, 21nfan 1898 . . 3 𝑦(𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)
2322nfab 2910 . 2 𝑦{𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
241, 23nfcxfr 2902 1 𝑦X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1537  wtru 1540  wnf 1782  wcel 2107  {cab 2713  wnfc 2889  wral 3060   Fn wfn 6555  cfv 6560  Xcixp 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2376  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-ixp 8939
This theorem is referenced by: (None)
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