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Theorem brwitnlem 7474
 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 6113 . . . . 5 (𝑂‘⟨𝐴, 𝐵⟩) ∈ V
2 dif1o 7467 . . . . 5 ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1𝑜) ↔ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ V ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
31, 2mpbiran 955 . . . 4 ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1𝑜) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
43anbi2i 726 . . 3 ((⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1𝑜)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
5 brwitnlem.o . . . 4 𝑂 Fn 𝑋
6 elpreima 6245 . . . 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 6128 . . . . . 6 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝑂 → (𝑂‘⟨𝐴, 𝐵⟩) = ∅)
98necon1ai 2809 . . . . 5 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom 𝑂)
10 fndm 5904 . . . . . 6 (𝑂 Fn 𝑋 → dom 𝑂 = 𝑋)
115, 10ax-mp 5 . . . . 5 dom 𝑂 = 𝑋
129, 11syl6eleq 2698 . . . 4 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑋)
1312pm4.71ri 663 . . 3 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
144, 7, 133bitr4i 291 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1𝑜)) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
15 brwitnlem.r . . . 4 𝑅 = (𝑂 “ (V ∖ 1𝑜))
1615breqi 4589 . . 3 (𝐴𝑅𝐵𝐴(𝑂 “ (V ∖ 1𝑜))𝐵)
17 df-br 4584 . . 3 (𝐴(𝑂 “ (V ∖ 1𝑜))𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1𝑜)))
1816, 17bitri 263 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1𝑜)))
19 df-ov 6552 . . 3 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
2019neeq1i 2846 . 2 ((𝐴𝑂𝐵) ≠ ∅ ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
2114, 18, 203bitr4i 291 1 (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037  dom cdm 5038   “ cima 5041   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549  1𝑜c1o 7440 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552  df-1o 7447 This theorem is referenced by:  brgic  17534  brric  18567  brlmic  18889  hmph  21389
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