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Theorem brwitnlem 8234
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 6730 . . . . 5 (𝑂‘⟨𝐴, 𝐵⟩) ∈ V
2 dif1o 8227 . . . . 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 626 . . 3 ((⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
5 brwitnlem.o . . . 4 𝑂 Fn 𝑋
6 elpreima 6878 . . . 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 6747 . . . . . 6 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝑂 → (𝑂‘⟨𝐴, 𝐵⟩) = ∅)
98necon1ai 2968 . . . . 5 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom 𝑂)
105fndmi 6482 . . . . 5 dom 𝑂 = 𝑋
119, 10eleqtrdi 2848 . . . 4 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑋)
1211pm4.71ri 564 . . 3 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
134, 7, 123bitr4i 306 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1o)) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
14 brwitnlem.r . . . 4 𝑅 = (𝑂 “ (V ∖ 1o))
1514breqi 5059 . . 3 (𝐴𝑅𝐵𝐴(𝑂 “ (V ∖ 1o))𝐵)
16 df-br 5054 . . 3 (𝐴(𝑂 “ (V ∖ 1o))𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1o)))
1715, 16bitri 278 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1o)))
18 df-ov 7216 . . 3 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
1918neeq1i 3005 . 2 ((𝐴𝑂𝐵) ≠ ∅ ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
2013, 17, 193bitr4i 306 1 (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wcel 2110  wne 2940  Vcvv 3408  cdif 3863  c0 4237  cop 4547   class class class wbr 5053  ccnv 5550  dom cdm 5551  cima 5554   Fn wfn 6375  cfv 6380  (class class class)co 7213  1oc1o 8195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388  df-ov 7216  df-1o 8202
This theorem is referenced by:  brgic  18673  brric  19764  brlmic  20105  hmph  22673
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