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Theorem rabxp 4407
Description: Membership in a class builder restricted to a cross product. (Contributed by NM, 20-Feb-2014.)
Hypothesis
Ref Expression
rabxp.1 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
Assertion
Ref Expression
rabxp {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵𝜓)}
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝜑,𝑦,𝑧   𝜓,𝑥
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦,𝑧)

Proof of Theorem rabxp
StepHypRef Expression
1 elxp 4389 . . . . 5 (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
21anbi1i 439 . . . 4 ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
3 19.41vv 1799 . . . 4 (∃𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
4 anass 387 . . . . . 6 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)))
5 rabxp.1 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
65anbi2d 445 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) ∧ 𝜑) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓)))
7 df-3an 898 . . . . . . . 8 ((𝑦𝐴𝑧𝐵𝜓) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓))
86, 7syl6bbr 191 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) ∧ 𝜑) ↔ (𝑦𝐴𝑧𝐵𝜓)))
98pm5.32i 435 . . . . . 6 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓)))
104, 9bitri 177 . . . . 5 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓)))
11102exbii 1513 . . . 4 (∃𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓)))
122, 3, 113bitr2i 201 . . 3 ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓)))
1312abbii 2169 . 2 {𝑥 ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {𝑥 ∣ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓))}
14 df-rab 2332 . 2 {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑)}
15 df-opab 3846 . 2 {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵𝜓)} = {𝑥 ∣ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓))}
1613, 14, 153eqtr4i 2086 1 {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵𝜓)}
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wex 1397  wcel 1409  {cab 2042  {crab 2327  cop 3405  {copab 3844   × cxp 4370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rab 2332  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-xp 4378
This theorem is referenced by: (None)
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