Theorem brcart 35208

Description: Binary relation form of the cartesian product operator. (Contributed by Scott Fenton, 11-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Hypotheses

Ref	Expression
brcart.1	⊢ 𝐴 ∈ V
brcart.2	⊢ 𝐵 ∈ V
brcart.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
brcart	⊢ (⟨𝐴, 𝐵⟩Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵))

Proof of Theorem brcart

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		opex 5463	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		brcart.3	. 2 ⊢ 𝐶 ∈ V
3		df-cart 35141	. 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 5715	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
7		brxp 5724	. . 3 ⊢ (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
8	6, 2, 7	mpbir2an 707	. 2 ⊢ ⟨𝐴, 𝐵⟩((V × V) × V)𝐶
9		3anass 1093	. . . . 5 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐵)))
10	4	epeli 5581	. . . . . . 7 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
11	5	epeli 5581	. . . . . . 7 ⊢ (𝑧 E 𝐵 ↔ 𝑧 ∈ 𝐵)
12	10, 11	anbi12i 625	. . . . . 6 ⊢ ((𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
13	12	anbi2i 621	. . . . 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 1849	. . 3 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
16		vex 3476	. . . 4 ⊢ 𝑥 ∈ V
17	16, 4, 5	brpprod3b 35163	. . 3 ⊢ (𝑥pprod( E , E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵))
18		elxp 5698	. . 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 35172	1 ⊢ (⟨𝐴, 𝐵⟩Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵))

Syntax hints:   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539  ∃wex 1779   ∈ wcel 2104  Vcvv 3472  ⟨cop 4633   class class class wbr 5147   E cep 5578   × cxp 5673  pprodcpprod 35107  Cartccart 35117

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-symdif 4241  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-eprel 5579  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fo 6548  df-fv 6550  df-1st 7977  df-2nd 7978  df-txp 35130  df-pprod 35131  df-cart 35141

This theorem is referenced by:  brimg  35213  brrestrict  35225