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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 8538 | . . . . 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 7078 | . . . 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 2968 | . . . . 5 ⊢ ((𝑂‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom 𝑂) |
| 10 | 5 | fndmi 6672 | . . . . 5 ⊢ dom 𝑂 = 𝑋 |
| 11 | 9, 10 | eleqtrdi 2851 | . . . 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 5149 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝑂 “ (V ∖ 1o))𝐵) |
| 16 | df-br 5144 | . . 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 7434 | . . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘〈𝐴, 𝐵〉) | |
| 19 | 18 | neeq1i 3005 | . 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 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∖ cdif 3948 ∅c0 4333 〈cop 4632 class class class wbr 5143 ◡ccnv 5684 dom cdm 5685 “ cima 5688 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 df-1o 8506 |
| This theorem is referenced by: brgic 19288 brric 20504 brlmic 21067 hmph 23784 brgric 47881 brgrlic 47964 |
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