Theorem brcart 33395

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 5358	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		brcart.3	. 2 ⊢ 𝐶 ∈ V
3		df-cart 33328	. 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 5596	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
7		brxp 5603	. . 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 5470	. . . . . . 7 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
11	5	epeli 5470	. . . . . . 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 1849	. . 3 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
16		vex 3499	. . . 4 ⊢ 𝑥 ∈ V
17	16, 4, 5	brpprod3b 33350	. . 3 ⊢ (𝑥pprod( E , E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵))
18		elxp 5580	. . 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 33359	1 ⊢ (⟨𝐴, 𝐵⟩Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵))

Syntax hints:   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3496  ⟨cop 4575   class class class wbr 5068   E cep 5466   × cxp 5555  pprodcpprod 33294  Cartccart 33304

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 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

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 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-symdif 4221  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-eprel 5467  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fo 6363  df-fv 6365  df-1st 7691  df-2nd 7692  df-txp 33317  df-pprod 33318  df-cart 33328

This theorem is referenced by:  brimg  33400  brrestrict  33412