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Theorem nfixpxy 6541
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 6523 . 2 X𝑥𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
2 nfcv 2240 . . . . 5 𝑦𝑧
3 nftru 1410 . . . . . . 7 𝑥
4 nfcvd 2241 . . . . . . . 8 (⊤ → 𝑦𝑥)
5 nfixp.1 . . . . . . . . 9 𝑦𝐴
65a1i 9 . . . . . . . 8 (⊤ → 𝑦𝐴)
74, 6nfeld 2256 . . . . . . 7 (⊤ → Ⅎ𝑦 𝑥𝐴)
83, 7nfabd 2259 . . . . . 6 (⊤ → 𝑦{𝑥𝑥𝐴})
98mptru 1308 . . . . 5 𝑦{𝑥𝑥𝐴}
102, 9nffn 5155 . . . 4 𝑦 𝑧 Fn {𝑥𝑥𝐴}
11 df-ral 2380 . . . . 5 (∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
122a1i 9 . . . . . . . . . 10 (⊤ → 𝑦𝑧)
1312, 4nffvd 5365 . . . . . . . . 9 (⊤ → 𝑦(𝑧𝑥))
14 nfixp.2 . . . . . . . . . 10 𝑦𝐵
1514a1i 9 . . . . . . . . 9 (⊤ → 𝑦𝐵)
1613, 15nfeld 2256 . . . . . . . 8 (⊤ → Ⅎ𝑦(𝑧𝑥) ∈ 𝐵)
177, 16nfimd 1532 . . . . . . 7 (⊤ → Ⅎ𝑦(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
183, 17nfald 1701 . . . . . 6 (⊤ → Ⅎ𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
1918mptru 1308 . . . . 5 𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵)
2011, 19nfxfr 1418 . . . 4 𝑦𝑥𝐴 (𝑧𝑥) ∈ 𝐵
2110, 20nfan 1512 . . 3 𝑦(𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)
2221nfab 2245 . 2 𝑦{𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
231, 22nfcxfr 2237 1 𝑦X𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1297  wtru 1300  wnf 1404  wcel 1448  {cab 2086  wnfc 2227  wral 2375   Fn wfn 5054  cfv 5059  Xcixp 6522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067  df-ixp 6523
This theorem is referenced by: (None)
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