Theorem brwitnlem 8520

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

Step	Hyp	Ref	Expression
1		fvex 6901	. . . . 5 ⊢ (𝑂‘⟨𝐴, 𝐵⟩) ∈ V
2		dif1o 8513	. . . . 5 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o) ↔ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ V ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
3	1, 2	mpbiran 722	. . . 4 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
4	3	anbi2i 635	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
5		brwitnlem.o	. . . 4 ⊢ 𝑂 Fn 𝑋
6		elpreima 7060	. . . 4 ⊢ (𝑂 Fn 𝑋 → (⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o))))
7	5, 6	ax-mp 5	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o)))
8		ndmfv 6920	. . . . . 6 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝑂 → (𝑂‘⟨𝐴, 𝐵⟩) = ∅)
9	8	necon1ai 2982	. . . . 5 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom 𝑂)
10	5	fndmi 6646	. . . . 5 ⊢ dom 𝑂 = 𝑋
11	9, 10	eleqtrdi 2870	. . . 4 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑋)
12	11	pm4.71ri 570	. . 3 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
13	4, 7, 12	3bitr4i 306	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o)) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
14		brwitnlem.r	. . . 4 ⊢ 𝑅 = (◡𝑂 “ (V ∖ 1o))
15	14	breqi 5108	. . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝑂 “ (V ∖ 1o))𝐵)
16		df-br 5103	. . 3 ⊢ (𝐴(◡𝑂 “ (V ∖ 1o))𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o)))
17	15, 16	bitri 278	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o)))
18		df-ov 7425	. . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
19	18	neeq1i 3019	. 2 ⊢ ((𝐴𝑂𝐵) ≠ ∅ ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
20	13, 17, 19	3bitr4i 306	1 ⊢ (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  Vcvv 3450   ∖ cdif 4044  ∅c0 4277  ⟨cop 4834   class class class wbr 5102  ^◡ccnv 5649  dom cdm 5650   “ cima 5653   Fn wfn 6536  ‘cfv 6541  (class class class)co 7422  1_oc1o 8474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-fv 6549  df-ov 7425  df-1o 8481

This theorem is used by:  brgic  19491  brric  20752  brlmic  21350  hmph  24102  brgric  49117  brgrlic  49209