Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 6685 | . . . . 5 ⊢ (𝑂‘〈𝐴, 𝐵〉) ∈ V | |
2 | dif1o 8127 | . . . . 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 6830 | . . . 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 6702 | . . . . . 6 ⊢ (¬ 〈𝐴, 𝐵〉 ∈ dom 𝑂 → (𝑂‘〈𝐴, 𝐵〉) = ∅) | |
9 | 8 | necon1ai 3045 | . . . . 5 ⊢ ((𝑂‘〈𝐴, 𝐵〉) ≠ ∅ → 〈𝐴, 𝐵〉 ∈ dom 𝑂) |
10 | fndm 6457 | . . . . . 6 ⊢ (𝑂 Fn 𝑋 → dom 𝑂 = 𝑋) | |
11 | 5, 10 | ax-mp 5 | . . . . 5 ⊢ dom 𝑂 = 𝑋 |
12 | 9, 11 | eleqtrdi 2925 | . . . 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 5074 | . . 3 ⊢ (𝐴𝑅𝐵 ↔ 𝐴(◡𝑂 “ (V ∖ 1o))𝐵) |
17 | df-br 5069 | . . 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 7161 | . . 3 ⊢ (𝐴𝑂𝐵) = (𝑂‘〈𝐴, 𝐵〉) | |
20 | 19 | neeq1i 3082 | . 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 1537 ∈ wcel 2114 ≠ wne 3018 Vcvv 3496 ∖ cdif 3935 ∅c0 4293 〈cop 4575 class class class wbr 5068 ◡ccnv 5556 dom cdm 5557 “ cima 5560 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 1oc1o 8097 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-fv 6365 df-ov 7161 df-1o 8104 |
This theorem is referenced by: brgic 18411 brric 19501 brlmic 19842 hmph 22386 |
Copyright terms: Public domain | W3C validator |