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Theorem nfixp 8081
Description: Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.)
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 8063 . 2 X𝑥𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
2 nfcv 2913 . . . . 5 𝑦𝑧
3 nftru 1878 . . . . . . 7 𝑥
4 nfcvf 2937 . . . . . . . . 9 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
54adantl 467 . . . . . . . 8 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝑥)
6 nfixp.1 . . . . . . . . 9 𝑦𝐴
76a1i 11 . . . . . . . 8 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝐴)
85, 7nfeld 2922 . . . . . . 7 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦 𝑥𝐴)
93, 8nfabd2 2933 . . . . . 6 (⊤ → 𝑦{𝑥𝑥𝐴})
109trud 1641 . . . . 5 𝑦{𝑥𝑥𝐴}
112, 10nffn 6127 . . . 4 𝑦 𝑧 Fn {𝑥𝑥𝐴}
12 df-ral 3066 . . . . 5 (∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
132a1i 11 . . . . . . . . . 10 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝑧)
1413, 5nffvd 6341 . . . . . . . . 9 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦(𝑧𝑥))
15 nfixp.2 . . . . . . . . . 10 𝑦𝐵
1615a1i 11 . . . . . . . . 9 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝐵)
1714, 16nfeld 2922 . . . . . . . 8 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑧𝑥) ∈ 𝐵)
188, 17nfimd 1973 . . . . . . 7 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
193, 18nfald2 2481 . . . . . 6 (⊤ → Ⅎ𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2019trud 1641 . . . . 5 𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵)
2112, 20nfxfr 1929 . . . 4 𝑦𝑥𝐴 (𝑧𝑥) ∈ 𝐵
2211, 21nfan 1980 . . 3 𝑦(𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)
2322nfab 2918 . 2 𝑦{𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
241, 23nfcxfr 2911 1 𝑦X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wal 1629  wtru 1632  wnf 1856  wcel 2145  {cab 2757  wnfc 2900  wral 3061   Fn wfn 6026  cfv 6031  Xcixp 8062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ixp 8063
This theorem is referenced by:  vonioo  41416
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