Theorem brwitnlem 7587

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 6201	. . . . 5 ⊢ (𝑂‘⟨𝐴, 𝐵⟩) ∈ V
2		dif1o 7580	. . . . 5 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1𝑜) ↔ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ V ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
3	1, 2	mpbiran 953	. . . 4 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1𝑜) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
4	3	anbi2i 730	. . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1𝑜)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
5		brwitnlem.o	. . . 4 ⊢ 𝑂 Fn 𝑋
6		elpreima 6337	. . . 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 6218	. . . . . 6 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝑂 → (𝑂‘⟨𝐴, 𝐵⟩) = ∅)
9	8	necon1ai 2821	. . . . 5 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom 𝑂)
10		fndm 5990	. . . . . 6 ⊢ (𝑂 Fn 𝑋 → dom 𝑂 = 𝑋)
11	5, 10	ax-mp 5	. . . . 5 ⊢ dom 𝑂 = 𝑋
12	9, 11	syl6eleq 2711	. . . 4 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑋)
13	12	pm4.71ri 665	. . 3 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅))
14	4, 7, 13	3bitr4i 292	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1𝑜)) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
15		brwitnlem.r	. . . 4 ⊢ 𝑅 = (◡𝑂 “ (V ∖ 1𝑜))
16	15	breqi 4659	. . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝑂 “ (V ∖ 1𝑜))𝐵)
17		df-br 4654	. . 3 ⊢ (𝐴(◡𝑂 “ (V ∖ 1𝑜))𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1𝑜)))
18	16, 17	bitri 264	. 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1𝑜)))
19		df-ov 6653	. . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
20	19	neeq1i 2858	. 2 ⊢ ((𝐴𝑂𝐵) ≠ ∅ ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)
21	14, 18, 20	3bitr4i 292	1 ⊢ (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅)

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  Vcvv 3200   ∖ cdif 3571  ∅c0 3915  ⟨cop 4183   class class class wbr 4653  ^◡ccnv 5113  dom cdm 5114   “ cima 5117   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-1o 7560

This theorem is referenced by:  brgic  17711  brric  18744  brlmic  19068  hmph  21579