![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > brcart | Structured version Visualization version GIF version |
Description: Binary relation form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
brcart.1 | ⊢ 𝐴 ∈ V |
brcart.2 | ⊢ 𝐵 ∈ V |
brcart.3 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
brcart | ⊢ (〈𝐴, 𝐵〉Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opex 5419 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
2 | brcart.3 | . 2 ⊢ 𝐶 ∈ V | |
3 | df-cart 34417 | . 2 ⊢ Cart = (((V × V) × V) ∖ ran ((V ⊗ E ) △ (pprod( E , E ) ⊗ V))) | |
4 | brcart.1 | . . . 4 ⊢ 𝐴 ∈ V | |
5 | brcart.2 | . . . 4 ⊢ 𝐵 ∈ V | |
6 | 4, 5 | opelvv 5670 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
7 | brxp 5679 | . . 3 ⊢ (〈𝐴, 𝐵〉((V × V) × V)𝐶 ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ 𝐶 ∈ V)) | |
8 | 6, 2, 7 | mpbir2an 709 | . 2 ⊢ 〈𝐴, 𝐵〉((V × V) × V)𝐶 |
9 | 3anass 1095 | . . . . 5 ⊢ ((𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐵))) | |
10 | 4 | epeli 5537 | . . . . . . 7 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴) |
11 | 5 | epeli 5537 | . . . . . . 7 ⊢ (𝑧 E 𝐵 ↔ 𝑧 ∈ 𝐵) |
12 | 10, 11 | anbi12i 627 | . . . . . 6 ⊢ ((𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) |
13 | 12 | anbi2i 623 | . . . . 5 ⊢ ((𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐵)) ↔ (𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) |
14 | 9, 13 | bitri 274 | . . . 4 ⊢ ((𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) |
15 | 14 | 2exbii 1851 | . . 3 ⊢ (∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) |
16 | vex 3447 | . . . 4 ⊢ 𝑥 ∈ V | |
17 | 16, 4, 5 | brpprod3b 34439 | . . 3 ⊢ (𝑥pprod( E , E )〈𝐴, 𝐵〉 ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵)) |
18 | elxp 5654 | . . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) | |
19 | 15, 17, 18 | 3bitr4ri 303 | . 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ 𝑥pprod( E , E )〈𝐴, 𝐵〉) |
20 | 1, 2, 3, 8, 19 | brtxpsd3 34448 | 1 ⊢ (〈𝐴, 𝐵〉Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3443 〈cop 4590 class class class wbr 5103 E cep 5534 × cxp 5629 pprodcpprod 34383 Cartccart 34393 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-symdif 4200 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-eprel 5535 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fo 6499 df-fv 6501 df-1st 7917 df-2nd 7918 df-txp 34406 df-pprod 34407 df-cart 34417 |
This theorem is referenced by: brimg 34489 brrestrict 34501 |
Copyright terms: Public domain | W3C validator |