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Theorem brpprod3a 33233
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 33231 . . . . . . 7 pprod(𝑅, 𝑆) ⊆ ((V × V) × (V × V))
21brel 5615 . . . . . 6 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ 𝑍 ∈ (V × V)))
32simprd 496 . . . . 5 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍𝑍 ∈ (V × V))
4 elvv 5624 . . . . 5 (𝑍 ∈ (V × V) ↔ ∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩)
53, 4sylib 219 . . . 4 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → ∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩)
65pm4.71ri 561 . . 3 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ (∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
7 19.41vv 1944 . . 3 (∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ (∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
86, 7bitr4i 279 . 2 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
9 breq2 5066 . . . 4 (𝑍 = ⟨𝑧, 𝑤⟩ → (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
109pm5.32i 575 . . 3 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
11102exbii 1842 . 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 3502 . . . . . 6 𝑧 ∈ V
15 vex 3502 . . . . . 6 𝑤 ∈ V
1612, 13, 14, 15brpprod 33232 . . . . 5 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩ ↔ (𝑋𝑅𝑧𝑌𝑆𝑤))
1716anbi2i 622 . . . 4 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ (𝑋𝑅𝑧𝑌𝑆𝑤)))
18 3anass 1089 . . . 4 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ (𝑋𝑅𝑧𝑌𝑆𝑤)))
1917, 18bitr4i 279 . . 3 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
20192exbii 1842 . 2 (∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
218, 11, 203bitri 298 1 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  Vcvv 3499  cop 4569   class class class wbr 5062   × cxp 5551  pprodcpprod 33178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fo 6357  df-fv 6359  df-1st 7683  df-2nd 7684  df-txp 33201  df-pprod 33202
This theorem is referenced by:  brpprod3b  33234  brapply  33285  dfrdg4  33298
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