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Theorem rabxp 4380
 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 4362 . . . . 5 (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)))
21anbi1i 431 . . . 4 ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
3 19.41vv 1783 . . . 4 (∃𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑))
4 anass 381 . . . . . 6 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)))
5 rabxp.1 . . . . . . . . 9 (𝑥 = ⟨𝑦, 𝑧⟩ → (𝜑𝜓))
65anbi2d 437 . . . . . . . 8 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) ∧ 𝜑) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓)))
7 df-3an 887 . . . . . . . 8 ((𝑦𝐴𝑧𝐵𝜓) ↔ ((𝑦𝐴𝑧𝐵) ∧ 𝜓))
86, 7syl6bbr 187 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → (((𝑦𝐴𝑧𝐵) ∧ 𝜑) ↔ (𝑦𝐴𝑧𝐵𝜓)))
98pm5.32i 427 . . . . . 6 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ ((𝑦𝐴𝑧𝐵) ∧ 𝜑)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓)))
104, 9bitri 173 . . . . 5 (((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓)))
11102exbii 1497 . . . 4 (∃𝑦𝑧((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵)) ∧ 𝜑) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓)))
122, 3, 113bitr2i 197 . . 3 ((𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓)))
1312abbii 2153 . 2 {𝑥 ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {𝑥 ∣ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓))}
14 df-rab 2315 . 2 {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ (𝐴 × 𝐵) ∧ 𝜑)}
15 df-opab 3819 . 2 {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵𝜓)} = {𝑥 ∣ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦𝐴𝑧𝐵𝜓))}
1613, 14, 153eqtr4i 2070 1 {𝑥 ∈ (𝐴 × 𝐵) ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ (𝑦𝐴𝑧𝐵𝜓)}
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 885   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026  {crab 2310  ⟨cop 3378  {copab 3817   × cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351 This theorem is referenced by: (None)
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