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Theorem brwitnlem 8545
Description: Lemma for relations which assert the existence of a witness in a two-parameter set. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 23-Aug-2015.)
Hypotheses
Ref Expression
brwitnlem.r 𝑅 = (𝑂 “ (V ∖ 1o))
brwitnlem.o 𝑂 Fn 𝑋
Assertion
Ref Expression
brwitnlem (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)

Proof of Theorem brwitnlem
StepHypRef Expression
1 fvex 6919 . . . . 5 (𝑂‘⟨𝐴, 𝐵⟩) ∈ V
2 dif1o 8538 . . . . 5 ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o) ↔ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ V ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
31, 2mpbiran 709 . . . 4 ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
43anbi2i 623 . . 3 ((⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
5 brwitnlem.o . . . 4 𝑂 Fn 𝑋
6 elpreima 7078 . . . 4 (𝑂 Fn 𝑋 → (⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1o)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o))))
75, 6ax-mp 5 . . 3 (⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1o)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o)))
8 ndmfv 6941 . . . . . 6 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝑂 → (𝑂‘⟨𝐴, 𝐵⟩) = ∅)
98necon1ai 2968 . . . . 5 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom 𝑂)
105fndmi 6672 . . . . 5 dom 𝑂 = 𝑋
119, 10eleqtrdi 2851 . . . 4 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑋)
1211pm4.71ri 560 . . 3 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
134, 7, 123bitr4i 303 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1o)) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
14 brwitnlem.r . . . 4 𝑅 = (𝑂 “ (V ∖ 1o))
1514breqi 5149 . . 3 (𝐴𝑅𝐵𝐴(𝑂 “ (V ∖ 1o))𝐵)
16 df-br 5144 . . 3 (𝐴(𝑂 “ (V ∖ 1o))𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1o)))
1715, 16bitri 275 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1o)))
18 df-ov 7434 . . 3 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
1918neeq1i 3005 . 2 ((𝐴𝑂𝐵) ≠ ∅ ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
2013, 17, 193bitr4i 303 1 (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wne 2940  Vcvv 3480  cdif 3948  c0 4333  cop 4632   class class class wbr 5143  ccnv 5684  dom cdm 5685  cima 5688   Fn wfn 6556  cfv 6561  (class class class)co 7431  1oc1o 8499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569  df-ov 7434  df-1o 8506
This theorem is referenced by:  brgic  19288  brric  20504  brlmic  21067  hmph  23784  brgric  47881  brgrlic  47964
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