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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 35886 | . . . . . . 7 ⊢ pprod(𝑅, 𝑆) ⊆ ((V × V) × (V × V)) | |
| 2 | 1 | brel 5749 | . . . . . 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 5759 | . . . . 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 1949 | . . 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 5146 | . . . 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 1848 | . 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 3483 | . . . . . 6 ⊢ 𝑧 ∈ V | |
| 15 | vex 3483 | . . . . . 6 ⊢ 𝑤 ∈ V | |
| 16 | 12, 13, 14, 15 | brpprod 35887 | . . . . 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 1848 | . 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 1539 ∃wex 1778 ∈ wcel 2107 Vcvv 3479 〈cop 4631 class class class wbr 5142 × cxp 5682 pprodcpprod 35833 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-1st 8015 df-2nd 8016 df-txp 35856 df-pprod 35857 | 
| This theorem is referenced by: brpprod3b 35889 brapply 35940 dfrdg4 35953 | 
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