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Theorem brtxp2 34853
Description: The binary relation over a tail cross when the second argument is not an ordered pair. (Contributed by Scott Fenton, 14-Apr-2014.) (Revised by Mario Carneiro, 3-May-2015.)
Hypothesis
Ref Expression
brtxp2.1 𝐴 ∈ V
Assertion
Ref Expression
brtxp2 (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Proof of Theorem brtxp2
StepHypRef Expression
1 txpss3v 34850 . . . . . . 7 (𝑅𝑆) ⊆ (V × (V × V))
21brel 5742 . . . . . 6 (𝐴(𝑅𝑆)𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ (V × V)))
32simprd 497 . . . . 5 (𝐴(𝑅𝑆)𝐵𝐵 ∈ (V × V))
4 elvv 5751 . . . . 5 (𝐵 ∈ (V × V) ↔ ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
53, 4sylib 217 . . . 4 (𝐴(𝑅𝑆)𝐵 → ∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩)
65pm4.71ri 562 . . 3 (𝐴(𝑅𝑆)𝐵 ↔ (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵))
7 19.41vv 1955 . . 3 (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ (∃𝑥𝑦 𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵))
86, 7bitr4i 278 . 2 (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵))
9 breq2 5153 . . . 4 (𝐵 = ⟨𝑥, 𝑦⟩ → (𝐴(𝑅𝑆)𝐵𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
109pm5.32i 576 . . 3 ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
11102exbii 1852 . 2 (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)𝐵) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩))
12 brtxp2.1 . . . . . 6 𝐴 ∈ V
13 vex 3479 . . . . . 6 𝑥 ∈ V
14 vex 3479 . . . . . 6 𝑦 ∈ V
1512, 13, 14brtxp 34852 . . . . 5 (𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩ ↔ (𝐴𝑅𝑥𝐴𝑆𝑦))
1615anbi2i 624 . . . 4 ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥𝐴𝑆𝑦)))
17 3anass 1096 . . . 4 ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ (𝐴𝑅𝑥𝐴𝑆𝑦)))
1816, 17bitr4i 278 . . 3 ((𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ (𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦))
19182exbii 1852 . 2 (∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴(𝑅𝑆)⟨𝑥, 𝑦⟩) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦))
208, 11, 193bitri 297 1 (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  Vcvv 3475  cop 4635   class class class wbr 5149   × cxp 5675  ctxp 34802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975  df-2nd 7976  df-txp 34826
This theorem is referenced by:  brsuccf  34913  brrestrict  34921
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