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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 5348 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
2 | brcart.3 | . 2 ⊢ 𝐶 ∈ V | |
3 | df-cart 33321 | . 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 5588 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
7 | brxp 5595 | . . 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 1091 | . . . . 5 ⊢ ((𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐵))) | |
10 | 4 | epeli 5462 | . . . . . . 7 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴) |
11 | 5 | epeli 5462 | . . . . . . 7 ⊢ (𝑧 E 𝐵 ↔ 𝑧 ∈ 𝐵) |
12 | 10, 11 | anbi12i 628 | . . . . . 6 ⊢ ((𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)) |
13 | 12 | anbi2i 624 | . . . . 5 ⊢ ((𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐵)) ↔ (𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) |
14 | 9, 13 | bitri 277 | . . . 4 ⊢ ((𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) |
15 | 14 | 2exbii 1845 | . . 3 ⊢ (∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) |
16 | vex 3497 | . . . 4 ⊢ 𝑥 ∈ V | |
17 | 16, 4, 5 | brpprod3b 33343 | . . 3 ⊢ (𝑥pprod( E , E )〈𝐴, 𝐵〉 ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵)) |
18 | elxp 5572 | . . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) | |
19 | 15, 17, 18 | 3bitr4ri 306 | . 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ 𝑥pprod( E , E )〈𝐴, 𝐵〉) |
20 | 1, 2, 3, 8, 19 | brtxpsd3 33352 | 1 ⊢ (〈𝐴, 𝐵〉Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 〈cop 4566 class class class wbr 5058 E cep 5458 × cxp 5547 pprodcpprod 33287 Cartccart 33297 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-symdif 4218 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-eprel 5459 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-1st 7683 df-2nd 7684 df-txp 33310 df-pprod 33311 df-cart 33321 |
This theorem is referenced by: brimg 33393 brrestrict 33405 |
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