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Mirrors > Home > MPE Home > Th. List > brwitnlem | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
brwitnlem.r | ⊢ 𝑅 = (◡𝑂 “ (V ∖ 1o)) |
brwitnlem.o | ⊢ 𝑂 Fn 𝑋 |
Ref | Expression |
---|---|
brwitnlem | ⊢ (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6904 | . . . . 5 ⊢ (𝑂‘⟨𝐴, 𝐵⟩) ∈ V | |
2 | dif1o 8499 | . . . . 5 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o) ↔ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ V ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)) | |
3 | 1, 2 | mpbiran 707 | . . . 4 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅) |
4 | 3 | anbi2i 623 | . . 3 ⊢ ((⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o)) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)) |
5 | brwitnlem.o | . . . 4 ⊢ 𝑂 Fn 𝑋 | |
6 | elpreima 7059 | . . . 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 6926 | . . . . . 6 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝑂 → (𝑂‘⟨𝐴, 𝐵⟩) = ∅) | |
9 | 8 | necon1ai 2968 | . . . . 5 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom 𝑂) |
10 | 5 | fndmi 6653 | . . . . 5 ⊢ dom 𝑂 = 𝑋 |
11 | 9, 10 | eleqtrdi 2843 | . . . 4 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑋) |
12 | 11 | pm4.71ri 561 | . . 3 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑋 ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)) |
13 | 4, 7, 12 | 3bitr4i 302 | . 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o)) ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅) |
14 | brwitnlem.r | . . . 4 ⊢ 𝑅 = (◡𝑂 “ (V ∖ 1o)) | |
15 | 14 | breqi 5154 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝑂 “ (V ∖ 1o))𝐵) |
16 | df-br 5149 | . . 3 ⊢ (𝐴(◡𝑂 “ (V ∖ 1o))𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o))) | |
17 | 15, 16 | bitri 274 | . 2 ⊢ (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (◡𝑂 “ (V ∖ 1o))) |
18 | df-ov 7411 | . . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩) | |
19 | 18 | neeq1i 3005 | . 2 ⊢ ((𝐴𝑂𝐵) ≠ ∅ ↔ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅) |
20 | 13, 17, 19 | 3bitr4i 302 | 1 ⊢ (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 Vcvv 3474 ∖ cdif 3945 ∅c0 4322 ⟨cop 4634 class class class wbr 5148 ◡ccnv 5675 dom cdm 5676 “ cima 5679 Fn wfn 6538 ‘cfv 6543 (class class class)co 7408 1oc1o 8458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 df-ov 7411 df-1o 8465 |
This theorem is referenced by: brgic 19142 brric 20282 brlmic 20678 hmph 23279 |
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