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Theorem nfixp 8332
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 8314 . 2 X𝑥𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
2 nfcv 2948 . . . . 5 𝑦𝑧
3 nftru 1787 . . . . . . 7 𝑥
4 nfcvf 2974 . . . . . . . . 9 (¬ ∀𝑦 𝑦 = 𝑥𝑦𝑥)
54adantl 482 . . . . . . . 8 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝑥)
6 nfixp.1 . . . . . . . . 9 𝑦𝐴
76a1i 11 . . . . . . . 8 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝐴)
85, 7nfeld 2957 . . . . . . 7 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦 𝑥𝐴)
93, 8nfabd2 2969 . . . . . 6 (⊤ → 𝑦{𝑥𝑥𝐴})
109mptru 1529 . . . . 5 𝑦{𝑥𝑥𝐴}
112, 10nffn 6325 . . . 4 𝑦 𝑧 Fn {𝑥𝑥𝐴}
12 df-ral 3109 . . . . 5 (∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
132a1i 11 . . . . . . . . . 10 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝑧)
1413, 5nffvd 6553 . . . . . . . . 9 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦(𝑧𝑥))
15 nfixp.2 . . . . . . . . . 10 𝑦𝐵
1615a1i 11 . . . . . . . . 9 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → 𝑦𝐵)
1714, 16nfeld 2957 . . . . . . . 8 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑧𝑥) ∈ 𝐵)
188, 17nfimd 1877 . . . . . . 7 ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
193, 18nfald2 2423 . . . . . 6 (⊤ → Ⅎ𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
2019mptru 1529 . . . . 5 𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵)
2112, 20nfxfr 1835 . . . 4 𝑦𝑥𝐴 (𝑧𝑥) ∈ 𝐵
2211, 21nfan 1882 . . 3 𝑦(𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)
2322nfab 2954 . 2 𝑦{𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
241, 23nfcxfr 2946 1 𝑦X𝑥𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wal 1520  wtru 1523  wnf 1766  wcel 2080  {cab 2774  wnfc 2932  wral 3104   Fn wfn 6223  cfv 6228  Xcixp 8313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-iota 6192  df-fun 6230  df-fn 6231  df-fv 6236  df-ixp 8314
This theorem is referenced by:  vonioo  42520
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