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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 6855 | . . . . 5 ⊢ (𝑂‘〈𝐴, 𝐵〉) ∈ V | |
2 | dif1o 8445 | . . . . 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 7008 | . . . 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 6877 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝑂 → (𝑂‘〈𝐴, 𝐵〉) = ∅) | |
9 | 8 | necon1ai 2971 | . . . . 5 ⊢ ((𝑂‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom 𝑂) |
10 | 5 | fndmi 6606 | . . . . 5 ⊢ dom 𝑂 = 𝑋 |
11 | 9, 10 | eleqtrdi 2848 | . . . 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 5111 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝑂 “ (V ∖ 1o))𝐵) |
16 | df-br 5106 | . . 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 7359 | . . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘〈𝐴, 𝐵〉) | |
19 | 18 | neeq1i 3008 | . 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 2943 Vcvv 3445 ∖ cdif 3907 ∅c0 4282 〈cop 4592 class class class wbr 5105 ◡ccnv 5632 dom cdm 5633 “ cima 5636 Fn wfn 6491 ‘cfv 6496 (class class class)co 7356 1oc1o 8404 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-fv 6504 df-ov 7359 df-1o 8411 |
This theorem is referenced by: brgic 19057 brric 20176 brlmic 20527 hmph 23125 |
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