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Theorem brwitnlem 8543
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 8536 . . . . 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 7077 . . . 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 2965 . . . . 5 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom 𝑂)
105fndmi 6672 . . . . 5 dom 𝑂 = 𝑋
119, 10eleqtrdi 2848 . . . 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 5153 . . 3 (𝐴𝑅𝐵𝐴(𝑂 “ (V ∖ 1o))𝐵)
16 df-br 5148 . . 3 (𝐴(𝑂 “ (V ∖ 1o))𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1o)))
1715, 16bitri 275 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1o)))
18 df-ov 7433 . . 3 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
1918neeq1i 3002 . 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 1536  wcel 2105  wne 2937  Vcvv 3477  cdif 3959  c0 4338  cop 4636   class class class wbr 5147  ccnv 5687  dom cdm 5688  cima 5691   Fn wfn 6557  cfv 6562  (class class class)co 7430  1oc1o 8497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-fv 6570  df-ov 7433  df-1o 8504
This theorem is referenced by:  brgic  19300  brric  20520  brlmic  21084  hmph  23799  brgric  47818  brgrlic  47899
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