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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 33458 | . . . . . . 7 ⊢ pprod(𝑅, 𝑆) ⊆ ((V × V) × (V × V)) | |
2 | 1 | brel 5581 | . . . . . 6 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → (〈𝑋, 𝑌〉 ∈ (V × V) ∧ 𝑍 ∈ (V × V))) |
3 | 2 | simprd 499 | . . . . 5 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → 𝑍 ∈ (V × V)) |
4 | elvv 5590 | . . . . 5 ⊢ (𝑍 ∈ (V × V) ↔ ∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉) | |
5 | 3, 4 | sylib 221 | . . . 4 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 → ∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉) |
6 | 5 | pm4.71ri 564 | . . 3 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ (∃𝑧∃𝑤 𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍)) |
7 | 19.41vv 1951 | . . 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 281 | . 2 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍)) |
9 | breq2 5034 | . . . 4 ⊢ (𝑍 = 〈𝑧, 𝑤〉 → (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉)) | |
10 | 9 | pm5.32i 578 | . . 3 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉)) |
11 | 10 | 2exbii 1850 | . 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 3444 | . . . . . 6 ⊢ 𝑧 ∈ V | |
15 | vex 3444 | . . . . . 6 ⊢ 𝑤 ∈ V | |
16 | 12, 13, 14, 15 | brpprod 33459 | . . . . 5 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉 ↔ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
17 | 16 | anbi2i 625 | . . . 4 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))) |
18 | 3anass 1092 | . . . 4 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ (𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤))) | |
19 | 17, 18 | bitr4i 281 | . . 3 ⊢ ((𝑍 = 〈𝑧, 𝑤〉 ∧ 〈𝑋, 𝑌〉pprod(𝑅, 𝑆)〈𝑧, 𝑤〉) ↔ (𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
20 | 19 | 2exbii 1850 | . 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 300 | 1 ⊢ (〈𝑋, 𝑌〉pprod(𝑅, 𝑆)𝑍 ↔ ∃𝑧∃𝑤(𝑍 = 〈𝑧, 𝑤〉 ∧ 𝑋𝑅𝑧 ∧ 𝑌𝑆𝑤)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 〈cop 4531 class class class wbr 5030 × cxp 5517 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: brpprod3b 33461 brapply 33512 dfrdg4 33525 |
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