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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 5469 | . 2 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
| 2 | brcart.3 | . 2 ⊢ 𝐶 ∈ V | |
| 3 | df-cart 35866 | . 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 5725 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ (V × V) |
| 7 | brxp 5734 | . . 3 ⊢ (〈𝐴, 𝐵〉((V × V) × V)𝐶 ↔ (〈𝐴, 𝐵〉 ∈ (V × V) ∧ 𝐶 ∈ V)) | |
| 8 | 6, 2, 7 | mpbir2an 711 | . 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 5586 | . . . . . . 7 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴) |
| 11 | 5 | epeli 5586 | . . . . . . 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 623 | . . . . 5 ⊢ ((𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐵)) ↔ (𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) |
| 14 | 9, 13 | bitri 275 | . . . 4 ⊢ ((𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) |
| 15 | 14 | 2exbii 1849 | . . 3 ⊢ (∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) |
| 16 | vex 3484 | . . . 4 ⊢ 𝑥 ∈ V | |
| 17 | 16, 4, 5 | brpprod3b 35888 | . . 3 ⊢ (𝑥pprod( E , E )〈𝐴, 𝐵〉 ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵)) |
| 18 | elxp 5708 | . . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = 〈𝑦, 𝑧〉 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))) | |
| 19 | 15, 17, 18 | 3bitr4ri 304 | . 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ 𝑥pprod( E , E )〈𝐴, 𝐵〉) |
| 20 | 1, 2, 3, 8, 19 | brtxpsd3 35897 | 1 ⊢ (〈𝐴, 𝐵〉Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 〈cop 4632 class class class wbr 5143 E cep 5583 × cxp 5683 pprodcpprod 35832 Cartccart 35842 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-symdif 4253 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-1st 8014 df-2nd 8015 df-txp 35855 df-pprod 35856 df-cart 35866 |
| This theorem is referenced by: brimg 35938 brrestrict 35950 |
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