![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > brpprod3a | Structured version Visualization version GIF version |
Description: Condition for parallel product when the last argument is not an ordered pair. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
brpprod3.1 | ⊢ 𝑋 ∈ V |
brpprod3.2 | ⊢ 𝑌 ∈ V |
brpprod3.3 | ⊢ 𝑍 ∈ V |
Ref | Expression |
---|---|
brpprod3a | ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pprodss4v 35866 | . . . . . . 7 ⊢ pprod(𝑅, 𝑆) ⊆ ((V × V) × (V × V)) | |
2 | 1 | brel 5754 | . . . . . 6 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → (〈𝑋, 𝑌〉 ∈ (V × V) ∧ 𝑍 ∈ (V × V))) |
3 | 2 | simprd 495 | . . . . 5 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → 𝑍 ∈ (V × V)) |
4 | elvv 5763 | . . . . 5 ⊢ (𝑍 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉) | |
5 | 3, 4 | sylib 218 | . . . 4 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → ∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉) |
6 | 5 | pm4.71ri 560 | . . 3 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ (∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍)) |
7 | 19.41vv 1948 | . . 3 ⊢ (∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍) ↔ (∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍)) | |
8 | 6, 7 | bitr4i 278 | . 2 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍)) |
9 | breq2 5152 | . . . 4 ⊢ (𝑍 = 〈𝑧, 𝑤〉 → (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉)) | |
10 | 9 | pm5.32i 574 | . . 3 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉)) |
11 | 10 | 2exbii 1846 | . 2 ⊢ (∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍) ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉)) |
12 | brpprod3.1 | . . . . . 6 ⊢ 𝑋 ∈ V | |
13 | brpprod3.2 | . . . . . 6 ⊢ 𝑌 ∈ V | |
14 | vex 3482 | . . . . . 6 ⊢ 𝑧 ∈ V | |
15 | vex 3482 | . . . . . 6 ⊢ 𝑤 ∈ V | |
16 | 12, 13, 14, 15 | brpprod 35867 | . . . . 5 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉 ↔ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
17 | 16 | anbi2i 623 | . . . 4 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))) |
18 | 3anass 1094 | . . . 4 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))) | |
19 | 17, 18 | bitr4i 278 | . . 3 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
20 | 19 | 2exbii 1846 | . 2 ⊢ (∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉) ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
21 | 8, 11, 20 | 3bitri 297 | 1 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 〈cop 4637 class class class wbr 5148 × cxp 5687 pprodcpprod 35813 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-1st 8013 df-2nd 8014 df-txp 35836 df-pprod 35837 |
This theorem is referenced by: brpprod3b 35869 brapply 35920 dfrdg4 35933 |
Copyright terms: Public domain | W3C validator |