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Theorem brpprod3a 35868
Description: Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Hypotheses
Ref Expression
brpprod3.1 𝑋 ∈ V
brpprod3.2 𝑌 ∈ V
brpprod3.3 𝑍 ∈ V
Assertion
Ref Expression
brpprod3a (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
Distinct variable groups:   𝑧,𝑤,𝑅   𝑤,𝑆,𝑧   𝑤,𝑋,𝑧   𝑤,𝑌,𝑧   𝑤,𝑍,𝑧

Proof of Theorem brpprod3a
StepHypRef Expression
1 pprodss4v 35866 . . . . . . 7 pprod(𝑅, 𝑆) ⊆ ((V × V) × (V × V))
21brel 5754 . . . . . 6 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ 𝑍 ∈ (V × V)))
32simprd 495 . . . . 5 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍𝑍 ∈ (V × V))
4 elvv 5763 . . . . 5 (𝑍 ∈ (V × V) ↔ ∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩)
53, 4sylib 218 . . . 4 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → ∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩)
65pm4.71ri 560 . . 3 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ (∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
7 19.41vv 1948 . . 3 (∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ (∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
86, 7bitr4i 278 . 2 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
9 breq2 5152 . . . 4 (𝑍 = ⟨𝑧, 𝑤⟩ → (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
109pm5.32i 574 . . 3 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
11102exbii 1846 . 2 (∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
12 brpprod3.1 . . . . . 6 𝑋 ∈ V
13 brpprod3.2 . . . . . 6 𝑌 ∈ V
14 vex 3482 . . . . . 6 𝑧 ∈ V
15 vex 3482 . . . . . 6 𝑤 ∈ V
1612, 13, 14, 15brpprod 35867 . . . . 5 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩ ↔ (𝑋𝑅𝑧𝑌𝑆𝑤))
1716anbi2i 623 . . . 4 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ (𝑋𝑅𝑧𝑌𝑆𝑤)))
18 3anass 1094 . . . 4 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ (𝑋𝑅𝑧𝑌𝑆𝑤)))
1917, 18bitr4i 278 . . 3 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
20192exbii 1846 . 2 (∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
218, 11, 203bitri 297 1 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cop 4637   class class class wbr 5148   × cxp 5687  pprodcpprod 35813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-pprod 35837
This theorem is referenced by:  brpprod3b  35869  brapply  35920  dfrdg4  35933
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