Theorem brcart 35974

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 5402	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		brcart.3	. 2 ⊢ 𝐶 ∈ V
3		df-cart 35907	. 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 5654	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
7		brxp 5663	. . 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 1094	. . . . 5 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐵)))
10	4	epeli 5516	. . . . . . 7 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
11	5	epeli 5516	. . . . . . 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 1850	. . 3 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
16		vex 3440	. . . 4 ⊢ 𝑥 ∈ V
17	16, 4, 5	brpprod3b 35929	. . 3 ⊢ (𝑥pprod( E , E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵))
18		elxp 5637	. . 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 35938	1 ⊢ (⟨𝐴, 𝐵⟩Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436  ⟨cop 4579   class class class wbr 5089   E cep 5513   × cxp 5612  pprodcpprod 35873  Cartccart 35883

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-symdif 4200  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-eprel 5514  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fo 6487  df-fv 6489  df-1st 7921  df-2nd 7922  df-txp 35896  df-pprod 35897  df-cart 35907

This theorem is referenced by:  brimg  35979  brrestrict  35993