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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 6919 | . . . . 5 ⊢ (𝑂‘〈𝐴, 𝐵〉) ∈ V | |
2 | dif1o 8536 | . . . . 5 ⊢ ((𝑂‘〈𝐴, 𝐵〉) ∈ (V ∖ 1o) ↔ ((𝑂‘〈𝐴, 𝐵〉) ∈ V ∧ (𝑂‘〈𝐴, 𝐵〉) ≠ ∅)) | |
3 | 1, 2 | mpbiran 709 | . . . 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 7077 | . . . 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 6941 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝑂 → (𝑂‘〈𝐴, 𝐵〉) = ∅) | |
9 | 8 | necon1ai 2965 | . . . . 5 ⊢ ((𝑂‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom 𝑂) |
10 | 5 | fndmi 6672 | . . . . 5 ⊢ dom 𝑂 = 𝑋 |
11 | 9, 10 | eleqtrdi 2848 | . . . 4 ⊢ ((𝑂‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ 𝑋) |
12 | 11 | pm4.71ri 560 | . . 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 5153 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝑂 “ (V ∖ 1o))𝐵) |
16 | df-br 5148 | . . 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 7433 | . . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘〈𝐴, 𝐵〉) | |
19 | 18 | neeq1i 3002 | . 2 ⊢ ((𝐴𝑂𝐵) ≠ ∅ ↔ (𝑂‘〈𝐴, 𝐵〉) ≠ ∅) |
20 | 13, 17, 19 | 3bitr4i 303 | 1 ⊢ (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 Vcvv 3477 ∖ cdif 3959 ∅c0 4338 〈cop 4636 class class class wbr 5147 ◡ccnv 5687 dom cdm 5688 “ cima 5691 Fn wfn 6557 ‘cfv 6562 (class class class)co 7430 1oc1o 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-fv 6570 df-ov 7433 df-1o 8504 |
This theorem is referenced by: brgic 19300 brric 20520 brlmic 21084 hmph 23799 brgric 47818 brgrlic 47899 |
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