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Theorem elaltxp 32057
Description: Membership in alternate Cartesian products. (Contributed by Scott Fenton, 23-Mar-2012.)
Assertion
Ref Expression
elaltxp (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦

Proof of Theorem elaltxp
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 elex 3207 . 2 (𝑋 ∈ (𝐴 ×× 𝐵) → 𝑋 ∈ V)
2 altopex 32042 . . . . 5 𝑥, 𝑦⟫ ∈ V
3 eleq1 2687 . . . . 5 (𝑋 = ⟪𝑥, 𝑦⟫ → (𝑋 ∈ V ↔ ⟪𝑥, 𝑦⟫ ∈ V))
42, 3mpbiri 248 . . . 4 (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
54a1i 11 . . 3 ((𝑥𝐴𝑦𝐵) → (𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V))
65rexlimivv 3032 . 2 (∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫ → 𝑋 ∈ V)
7 eqeq1 2624 . . . 4 (𝑧 = 𝑋 → (𝑧 = ⟪𝑥, 𝑦⟫ ↔ 𝑋 = ⟪𝑥, 𝑦⟫))
872rexbidv 3053 . . 3 (𝑧 = 𝑋 → (∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫ ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
9 df-altxp 32041 . . 3 (𝐴 ×× 𝐵) = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = ⟪𝑥, 𝑦⟫}
108, 9elab2g 3347 . 2 (𝑋 ∈ V → (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫))
111, 6, 10pm5.21nii 368 1 (𝑋 ∈ (𝐴 ×× 𝐵) ↔ ∃𝑥𝐴𝑦𝐵 𝑋 = ⟪𝑥, 𝑦⟫)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wrex 2910  Vcvv 3195  caltop 32038   ×× caltxp 32039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-un 3572  df-nul 3908  df-sn 4169  df-pr 4171  df-altop 32040  df-altxp 32041
This theorem is referenced by:  altopelaltxp  32058  altxpsspw  32059
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