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Theorem nfixpxy 6691
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 6673 . 2 X𝑥𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
2 nfcv 2312 . . . . 5 𝑦𝑧
3 nftru 1459 . . . . . . 7 𝑥
4 nfcvd 2313 . . . . . . . 8 (⊤ → 𝑦𝑥)
5 nfixp.1 . . . . . . . . 9 𝑦𝐴
65a1i 9 . . . . . . . 8 (⊤ → 𝑦𝐴)
74, 6nfeld 2328 . . . . . . 7 (⊤ → Ⅎ𝑦 𝑥𝐴)
83, 7nfabd 2332 . . . . . 6 (⊤ → 𝑦{𝑥𝑥𝐴})
98mptru 1357 . . . . 5 𝑦{𝑥𝑥𝐴}
102, 9nffn 5292 . . . 4 𝑦 𝑧 Fn {𝑥𝑥𝐴}
11 df-ral 2453 . . . . 5 (∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
122a1i 9 . . . . . . . . . 10 (⊤ → 𝑦𝑧)
1312, 4nffvd 5506 . . . . . . . . 9 (⊤ → 𝑦(𝑧𝑥))
14 nfixp.2 . . . . . . . . . 10 𝑦𝐵
1514a1i 9 . . . . . . . . 9 (⊤ → 𝑦𝐵)
1613, 15nfeld 2328 . . . . . . . 8 (⊤ → Ⅎ𝑦(𝑧𝑥) ∈ 𝐵)
177, 16nfimd 1578 . . . . . . 7 (⊤ → Ⅎ𝑦(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
183, 17nfald 1753 . . . . . 6 (⊤ → Ⅎ𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵))
1918mptru 1357 . . . . 5 𝑦𝑥(𝑥𝐴 → (𝑧𝑥) ∈ 𝐵)
2011, 19nfxfr 1467 . . . 4 𝑦𝑥𝐴 (𝑧𝑥) ∈ 𝐵
2110, 20nfan 1558 . . 3 𝑦(𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)
2221nfab 2317 . 2 𝑦{𝑧 ∣ (𝑧 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑧𝑥) ∈ 𝐵)}
231, 22nfcxfr 2309 1 𝑦X𝑥𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1346  wtru 1349  wnf 1453  wcel 2141  {cab 2156  wnfc 2299  wral 2448   Fn wfn 5191  cfv 5196  Xcixp 6672
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fn 5199  df-fv 5204  df-ixp 6673
This theorem is referenced by: (None)
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