Theorem brwitnlem 8507

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 6905	. . . . 5 ⊢ (𝑂‘⟨𝐴, 𝐵⟩) ∈ V
2		dif1o 8500	. . . . 5 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o) ↔ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ V ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
3	1, 2	mpbiran 708	. . . 4 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
4	3	anbi2i 624	. . 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 6927	. . . . . 6 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝑂 → (𝑂‘⟨𝐴, 𝐵⟩) = ∅)
9	8	necon1ai 2969	. . . . 5 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom 𝑂)
10	5	fndmi 6654	. . . . 5 ⊢ dom 𝑂 = 𝑋
11	9, 10	eleqtrdi 2844	. . . 4 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑋)
12	11	pm4.71ri 562	. . 3 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
13	4, 7, 12	3bitr4i 303	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o)) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
14		brwitnlem.r	. . . 4 ⊢ 𝑅 = (◡𝑂 “ (V ∖ 1o))
15	14	breqi 5155	. . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝑂 “ (V ∖ 1o))𝐵)
16		df-br 5150	. . 3 ⊢ (𝐴(◡𝑂 “ (V ∖ 1o))𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o)))
17	15, 16	bitri 275	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o)))
18		df-ov 7412	. . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
19	18	neeq1i 3006	. 2 ⊢ ((𝐴𝑂𝐵) ≠ ∅ ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
20	13, 17, 19	3bitr4i 303	1 ⊢ (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)

Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   ≠ wne 2941  Vcvv 3475   ∖ cdif 3946  ∅c0 4323  ⟨cop 4635   class class class wbr 5149  ^◡ccnv 5676  dom cdm 5677   “ cima 5680   Fn wfn 6539  ‘cfv 6544  (class class class)co 7409  1_oc1o 8459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-ov 7412  df-1o 8466

This theorem is referenced by:  brgic  19143  brric  20283  brlmic  20679  hmph  23280