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Theorem brpprod3a 31688
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 31686 . . . . . . 7 pprod(𝑅, 𝑆) ⊆ ((V × V) × (V × V))
21brel 5138 . . . . . 6 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → (⟨𝑋, 𝑌⟩ ∈ (V × V) ∧ 𝑍 ∈ (V × V)))
32simprd 479 . . . . 5 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍𝑍 ∈ (V × V))
4 elvv 5148 . . . . 5 (𝑍 ∈ (V × V) ↔ ∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩)
53, 4sylib 208 . . . 4 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 → ∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩)
65pm4.71ri 664 . . 3 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ (∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
7 19.41vv 1912 . . 3 (∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ (∃𝑧𝑤 𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
86, 7bitr4i 267 . 2 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍))
9 breq2 4627 . . . 4 (𝑍 = ⟨𝑧, 𝑤⟩ → (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
109pm5.32i 668 . . 3 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩))
11102exbii 1772 . 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 3193 . . . . . 6 𝑧 ∈ V
15 vex 3193 . . . . . 6 𝑤 ∈ V
1612, 13, 14, 15brpprod 31687 . . . . 5 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩ ↔ (𝑋𝑅𝑧𝑌𝑆𝑤))
1716anbi2i 729 . . . 4 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ (𝑋𝑅𝑧𝑌𝑆𝑤)))
18 3anass 1040 . . . 4 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ (𝑋𝑅𝑧𝑌𝑆𝑤)))
1917, 18bitr4i 267 . . 3 ((𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ (𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
20192exbii 1772 . 2 (∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ ⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)⟨𝑧, 𝑤⟩) ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
218, 11, 203bitri 286 1 (⟨𝑋, 𝑌⟩pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧𝑤(𝑍 = ⟨𝑧, 𝑤⟩ ∧ 𝑋𝑅𝑧𝑌𝑆𝑤))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  Vcvv 3190  cop 4161   class class class wbr 4623   × cxp 5082  pprodcpprod 31632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-1st 7128  df-2nd 7129  df-txp 31655  df-pprod 31656
This theorem is referenced by:  brpprod3b  31689  brapply  31740  dfrdg4  31753
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