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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 6678 | . . . . 5 ⊢ (𝑂‘〈𝐴, 𝐵〉) ∈ V | |
2 | dif1o 8119 | . . . . 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 624 | . . 3 ⊢ ((〈𝐴, 𝐵〉 ∈ 𝑋 ∧ (𝑂‘〈𝐴, 𝐵〉) ∈ (V ∖ 1o)) ↔ (〈𝐴, 𝐵〉 ∈ 𝑋 ∧ (𝑂‘〈𝐴, 𝐵〉) ≠ ∅)) |
5 | brwitnlem.o | . . . 4 ⊢ 𝑂 Fn 𝑋 | |
6 | elpreima 6823 | . . . 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 6695 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝑂 → (𝑂‘〈𝐴, 𝐵〉) = ∅) | |
9 | 8 | necon1ai 3043 | . . . . 5 ⊢ ((𝑂‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom 𝑂) |
10 | fndm 6450 | . . . . . 6 ⊢ (𝑂 Fn 𝑋 → dom 𝑂 = 𝑋) | |
11 | 5, 10 | ax-mp 5 | . . . . 5 ⊢ dom 𝑂 = 𝑋 |
12 | 9, 11 | eleqtrdi 2923 | . . . 4 ⊢ ((𝑂‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ 𝑋) |
13 | 12 | pm4.71ri 563 | . . 3 ⊢ ((𝑂‘〈𝐴, 𝐵〉) ≠ ∅ ↔ (〈𝐴, 𝐵〉 ∈ 𝑋 ∧ (𝑂‘〈𝐴, 𝐵〉) ≠ ∅)) |
14 | 4, 7, 13 | 3bitr4i 305 | . 2 ⊢ (〈𝐴, 𝐵〉 ∈ (◡𝑂 “ (V ∖ 1o)) ↔ (𝑂‘〈𝐴, 𝐵〉) ≠ ∅) |
15 | brwitnlem.r | . . . 4 ⊢ 𝑅 = (◡𝑂 “ (V ∖ 1o)) | |
16 | 15 | breqi 5065 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝑂 “ (V ∖ 1o))𝐵) |
17 | df-br 5060 | . . 3 ⊢ (𝐴(◡𝑂 “ (V ∖ 1o))𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝑂 “ (V ∖ 1o))) | |
18 | 16, 17 | bitri 277 | . 2 ⊢ (𝐴𝑅𝐵 ↔ 〈𝐴, 𝐵〉 ∈ (◡𝑂 “ (V ∖ 1o))) |
19 | df-ov 7153 | . . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘〈𝐴, 𝐵〉) | |
20 | 19 | neeq1i 3080 | . 2 ⊢ ((𝐴𝑂𝐵) ≠ ∅ ↔ (𝑂‘〈𝐴, 𝐵〉) ≠ ∅) |
21 | 14, 18, 20 | 3bitr4i 305 | 1 ⊢ (𝐴𝑅𝐵 ↔ (𝐴𝑂𝐵) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3495 ∖ cdif 3933 ∅c0 4291 〈cop 4567 class class class wbr 5059 ◡ccnv 5549 dom cdm 5550 “ cima 5553 Fn wfn 6345 ‘cfv 6350 (class class class)co 7150 1oc1o 8089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-fv 6358 df-ov 7153 df-1o 8096 |
This theorem is referenced by: brgic 18403 brric 19493 brlmic 19834 hmph 22378 |
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