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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 6859 | . . . . 5 ⊢ (𝑂‘⟨𝐴, 𝐵⟩) ∈ V | |
2 | dif1o 8450 | . . . . 5 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ (V ∖ 1o) ↔ ((𝑂‘⟨𝐴, 𝐵⟩) ∈ V ∧ (𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅)) | |
3 | 1, 2 | mpbiran 708 | . . . 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 7012 | . . . 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 6881 | . . . . . 6 ⊢ (¬ ⟨𝐴, 𝐵⟩ ∈ dom 𝑂 → (𝑂‘⟨𝐴, 𝐵⟩) = ∅) | |
9 | 8 | necon1ai 2968 | . . . . 5 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ dom 𝑂) |
10 | 5 | fndmi 6610 | . . . . 5 ⊢ dom 𝑂 = 𝑋 |
11 | 9, 10 | eleqtrdi 2844 | . . . 4 ⊢ ((𝑂‘⟨𝐴, 𝐵⟩) ≠ ∅ → ⟨𝐴, 𝐵⟩ ∈ 𝑋) |
12 | 11 | pm4.71ri 562 | . . 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 5115 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝑂 “ (V ∖ 1o))𝐵) |
16 | df-br 5110 | . . 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 7364 | . . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 Vcvv 3447 ∖ cdif 3911 ∅c0 4286 ⟨cop 4596 class class class wbr 5109 ◡ccnv 5636 dom cdm 5637 “ cima 5640 Fn wfn 6495 ‘cfv 6500 (class class class)co 7361 1oc1o 8409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-fv 6508 df-ov 7364 df-1o 8416 |
This theorem is referenced by: brgic 19067 brric 20188 brlmic 20573 hmph 23150 |
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