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Theorem brwitnlem 7547
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 6168 . . . . 5 (𝑂‘⟨𝐴, 𝐵⟩) ∈ V
2 dif1o 7540 . . . . 5 ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1𝑜) ↔ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ V ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
31, 2mpbiran 952 . . . 4 ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1𝑜) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
43anbi2i 729 . . 3 ((⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1𝑜)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
5 brwitnlem.o . . . 4 𝑂 Fn 𝑋
6 elpreima 6303 . . . 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 6185 . . . . . 6 (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝑂 → (𝑂‘⟨𝐴, 𝐵⟩) = ∅)
98necon1ai 2817 . . . . 5 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom 𝑂)
10 fndm 5958 . . . . . 6 (𝑂 Fn 𝑋 → dom 𝑂 = 𝑋)
115, 10ax-mp 5 . . . . 5 dom 𝑂 = 𝑋
129, 11syl6eleq 2708 . . . 4 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑋)
1312pm4.71ri 664 . . 3 ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
144, 7, 133bitr4i 292 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1𝑜)) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
15 brwitnlem.r . . . 4 𝑅 = (𝑂 “ (V ∖ 1𝑜))
1615breqi 4629 . . 3 (𝐴𝑅𝐵𝐴(𝑂 “ (V ∖ 1𝑜))𝐵)
17 df-br 4624 . . 3 (𝐴(𝑂 “ (V ∖ 1𝑜))𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1𝑜)))
1816, 17bitri 264 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑂 “ (V ∖ 1𝑜)))
19 df-ov 6618 . . 3 (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
2019neeq1i 2854 . 2 ((𝐴𝑂𝐵) ≠ ∅ ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
2114, 18, 203bitr4i 292 1 (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3190  cdif 3557  c0 3897  cop 4161   class class class wbr 4623  ccnv 5083  dom cdm 5084  cima 5087   Fn wfn 5852  cfv 5857  (class class class)co 6615  1𝑜c1o 7513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865  df-ov 6618  df-1o 7520
This theorem is referenced by:  brgic  17651  brric  18684  brlmic  19008  hmph  21519
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