| 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 35884 | . . 3 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆) | |
| 2 | 1 | breqi 5149 | . 2 ⊢ (𝑋pprod(𝑅, 𝑆)〈𝑌, 𝑍〉 ↔ 𝑋◡pprod(◡𝑅, ◡𝑆)〈𝑌, 𝑍〉) |
| 3 | brpprod3.1 | . . . . 5 ⊢ 𝑋 ∈ V | |
| 4 | opex 5469 | . . . . 5 ⊢ 〈𝑌, 𝑍〉 ∈ V | |
| 5 | 3, 4 | brcnv 5893 | . . . 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 35887 | . . . 4 ⊢ (〈𝑌, 𝑍〉pprod(◡𝑅, ◡𝑆)𝑋 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤)) |
| 9 | 5, 8 | bitri 275 | . . 3 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤)) |
| 10 | biid 261 | . . . . 5 ⊢ (𝑋 = 〈𝑧, 𝑤〉 ↔ 𝑋 = 〈𝑧, 𝑤〉) | |
| 11 | vex 3484 | . . . . . 6 ⊢ 𝑧 ∈ V | |
| 12 | 6, 11 | brcnv 5893 | . . . . 5 ⊢ (𝑌◡𝑅𝑧 ↔ 𝑧𝑅𝑌) |
| 13 | vex 3484 | . . . . . 6 ⊢ 𝑤 ∈ V | |
| 14 | 7, 13 | brcnv 5893 | . . . . 5 ⊢ (𝑍◡𝑆𝑤 ↔ 𝑤𝑆𝑍) |
| 15 | 10, 12, 14 | 3anbi123i 1156 | . . . 4 ⊢ ((𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ (𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
| 16 | 15 | 2exbii 1849 | . . 3 ⊢ (∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤) ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
| 17 | 9, 16 | bitri 275 | . 2 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
| 18 | 2, 17 | bitri 275 | 1 ⊢ (𝑋pprod(𝑅, 𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 〈cop 4632 class class class wbr 5143 ◡ccnv 5684 pprodcpprod 35832 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 df-pprod 35856 |
| This theorem is referenced by: brcart 35933 |
| Copyright terms: Public domain | W3C validator |