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Theorem nfixpxy 6618
Description: Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.)
Hypotheses
Ref Expression
nfixp.1 𝑦𝐴
nfixp.2 𝑦𝐵
Assertion
Ref Expression
nfixpxy 𝑦X𝑥𝐴 𝐵
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)

Proof of Theorem nfixpxy
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-ixp 6600 . 2 X𝑥𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
2 nfcv 2282 . . . . 5 𝑦𝑧
3 nftru 1443 . . . . . . 7 𝑥
4 nfcvd 2283 . . . . . . . 8 (⊤ → 𝑦𝑥)
5 nfixp.1 . . . . . . . . 9 𝑦𝐴
65a1i 9 . . . . . . . 8 (⊤ → 𝑦𝐴)
74, 6nfeld 2298 . . . . . . 7 (⊤ → Ⅎ𝑦 𝑥𝐴)
83, 7nfabd 2301 . . . . . 6 (⊤ → 𝑦{𝑥𝑥𝐴})
98mptru 1341 . . . . 5 𝑦{𝑥𝑥𝐴}
102, 9nffn 5226 . . . 4 𝑦 𝑧 Fn {𝑥𝑥𝐴}
11 df-ral 2422 . . . . 5 (∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
122a1i 9 . . . . . . . . . 10 (⊤ → 𝑦𝑧)
1312, 4nffvd 5440 . . . . . . . . 9 (⊤ → 𝑦(𝑧𝑥))
14 nfixp.2 . . . . . . . . . 10 𝑦𝐵
1514a1i 9 . . . . . . . . 9 (⊤ → 𝑦𝐵)
1613, 15nfeld 2298 . . . . . . . 8 (⊤ → Ⅎ𝑦(𝑧𝑥) ∈ 𝐵)
177, 16nfimd 1565 . . . . . . 7 (⊤ → Ⅎ𝑦(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
183, 17nfald 1734 . . . . . 6 (⊤ → Ⅎ𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
1918mptru 1341 . . . . 5 𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵)
2011, 19nfxfr 1451 . . . 4 𝑦𝑥𝐴 (𝑧𝑥) ∈ 𝐵
2110, 20nfan 1545 . . 3 𝑦(𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)
2221nfab 2287 . 2 𝑦{𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
231, 22nfcxfr 2279 1 𝑦X𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1330  wtru 1333  wnf 1437  wcel 1481  {cab 2126  wnfc 2269  wral 2417   Fn wfn 5125  cfv 5130  Xcixp 6599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fn 5133  df-fv 5138  df-ixp 6600
This theorem is referenced by: (None)
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