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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 33351 | . . 3 ⊢ pprod(𝑅, 𝑆) = ◡pprod(◡𝑅, ◡𝑆) | |
2 | 1 | breqi 5058 | . 2 ⊢ (𝑋pprod(𝑅, 𝑆)〈𝑌, 𝑍〉 ↔ 𝑋◡pprod(◡𝑅, ◡𝑆)〈𝑌, 𝑍〉) |
3 | brpprod3.1 | . . . . 5 ⊢ 𝑋 ∈ V | |
4 | opex 5342 | . . . . 5 ⊢ 〈𝑌, 𝑍〉 ∈ V | |
5 | 3, 4 | brcnv 5739 | . . . 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 33354 | . . . 4 ⊢ (〈𝑌, 𝑍〉pprod(◡𝑅, ◡𝑆)𝑋 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤)) |
9 | 5, 8 | bitri 277 | . . 3 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑌◡𝑅𝑧 ∧ 𝑍◡𝑆𝑤)) |
10 | biid 263 | . . . . 5 ⊢ (𝑋 = 〈𝑧, 𝑤〉 ↔ 𝑋 = 〈𝑧, 𝑤〉) | |
11 | vex 3489 | . . . . . 6 ⊢ 𝑧 ∈ V | |
12 | 6, 11 | brcnv 5739 | . . . . 5 ⊢ (𝑌◡𝑅𝑧 ↔ 𝑧𝑅𝑌) |
13 | vex 3489 | . . . . . 6 ⊢ 𝑤 ∈ V | |
14 | 7, 13 | brcnv 5739 | . . . . 5 ⊢ (𝑍◡𝑆𝑤 ↔ 𝑤𝑆𝑍) |
15 | 10, 12, 14 | 3anbi123i 1151 | . . . 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 277 | . 2 ⊢ (𝑋◡pprod(◡𝑅, ◡𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
18 | 2, 17 | bitri 277 | 1 ⊢ (𝑋pprod(𝑅, 𝑆)〈𝑌, 𝑍〉 ↔ ∃𝑧∃𝑤(𝑋 = 〈𝑧, 𝑤〉 ∧ 𝑧𝑅𝑌 ∧ 𝑤𝑆𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 Vcvv 3486 〈cop 4559 class class class wbr 5052 ◡ccnv 5540 pprodcpprod 33299 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fo 6347 df-fv 6349 df-1st 7675 df-2nd 7676 df-txp 33322 df-pprod 33323 |
This theorem is referenced by: brcart 33400 |
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