![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > brpprod3b | Structured version Visualization version GIF version |
Description: Condition for parallel product when the first argument is not an ordered pair. (Contributed by Scott Fenton, 3-May-2014.) |
Ref | Expression |
---|---|
brpprod3.1 | ⊢ 𝑋 ∈ V |
brpprod3.2 | ⊢ 𝑌 ∈ V |
brpprod3.3 | ⊢ 𝑍 ∈ V |
Ref | Expression |
---|---|
brpprod3b | ⊢ (𝑋pprod(𝑅, 𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pprodcnveq 33457 | . . 3 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆) | |
2 | 1 | breqi 5036 | . 2 ⊢ (𝑋pprod(𝑅, 𝑆)〈𝑌, 𝑍〉 ↔ 𝑋◡pprod(◡𝑅, ◡𝑆)〈𝑌, 𝑍〉) |
3 | brpprod3.1 | . . . . 5 ⊢ 𝑋 ∈ V | |
4 | opex 5321 | . . . . 5 ⊢ 〈𝑌, 𝑍〉 ∈ V | |
5 | 3, 4 | brcnv 5717 | . . . 4 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)〈𝑌, 𝑍〉 ↔ 〈𝑌, 𝑍〉pprod(◡𝑅, ◡𝑆)𝑋) |
6 | brpprod3.2 | . . . . 5 ⊢ 𝑌 ∈ V | |
7 | brpprod3.3 | . . . . 5 ⊢ 𝑍 ∈ V | |
8 | 6, 7, 3 | brpprod3a 33460 | . . . 4 ⊢ (〈𝑌, 𝑍〉pprod(◡𝑅, ◡𝑆)𝑋 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤)) |
9 | 5, 8 | bitri 278 | . . 3 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤)) |
10 | biid 264 | . . . . 5 ⊢ (𝑋 = 〈𝑧, 𝑤〉 ↔ 𝑋 = 〈𝑧, 𝑤〉) | |
11 | vex 3444 | . . . . . 6 ⊢ 𝑧 ∈ V | |
12 | 6, 11 | brcnv 5717 | . . . . 5 ⊢ (𝑌◡𝑅𝑧 ↔ 𝑧𝑅𝑌) |
13 | vex 3444 | . . . . . 6 ⊢ 𝑤 ∈ V | |
14 | 7, 13 | brcnv 5717 | . . . . 5 ⊢ (𝑍◡𝑆𝑤 ↔ 𝑤𝑆𝑍) |
15 | 10, 12, 14 | 3anbi123i 1152 | . . . 4 ⊢ ((𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ (𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
16 | 15 | 2exbii 1850 | . . 3 ⊢ (∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
17 | 9, 16 | bitri 278 | . 2 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
18 | 2, 17 | bitri 278 | 1 ⊢ (𝑋pprod(𝑅, 𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 〈cop 4531 class class class wbr 5030 ◡ccnv 5518 pprodcpprod 33405 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-1st 7671 df-2nd 7672 df-txp 33428 df-pprod 33429 |
This theorem is referenced by: brcart 33506 |
Copyright terms: Public domain | W3C validator |