Theorem brcart 36422

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 5445	. 2 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
2		brcart.3	. 2 ⊢ 𝐶 ∈ V
3		df-cart 36355	. 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 5701	. . 3 ⊢ ⟨𝐴, 𝐵⟩ ∈ (V × V)
7		brxp 5710	. . 3 ⊢ (⟨𝐴, 𝐵⟩((V × V) × V)𝐶 ↔ (⟨𝐴, 𝐵⟩ ∈ (V × V) ∧ 𝐶 ∈ V))
8	6, 2, 7	mpbir2an 723	. 2 ⊢ ⟨𝐴, 𝐵⟩((V × V) × V)𝐶
9		3anass 1111	. . . . 5 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐵)))
10	4	epeli 5563	. . . . . . 7 ⊢ (𝑦 E 𝐴 ↔ 𝑦 ∈ 𝐴)
11	5	epeli 5563	. . . . . . 7 ⊢ (𝑧 E 𝐵 ↔ 𝑧 ∈ 𝐵)
12	10, 11	anbi12i 639	. . . . . 6 ⊢ ((𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
13	12	anbi2i 634	. . . . 5 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 E 𝐴 ∧ 𝑧 E 𝐵)) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
14	9, 13	bitri 278	. . . 4 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
15	14	2exbii 1879	. . 3 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
16		vex 3459	. . . 4 ⊢ 𝑥 ∈ V
17	16, 4, 5	brpprod3b 36377	. . 3 ⊢ (𝑥pprod( E , E )⟨𝐴, 𝐵⟩ ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝑦 E 𝐴 ∧ 𝑧 E 𝐵))
18		elxp 5684	. . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵)))
19	15, 17, 18	3bitr4ri 307	. 2 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ 𝑥pprod( E , E )⟨𝐴, 𝐵⟩)
20	1, 2, 3, 8, 19	brtxpsd3 36386	1 ⊢ (⟨𝐴, 𝐵⟩Cart𝐶 ↔ 𝐶 = (𝐴 × 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  Vcvv 3455  ⟨cop 4595   class class class wbr 5109   E cep 5560   × cxp 5659  pprodcpprod 36321  Cartccart 36331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-symdif 4206  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7982  df-2nd 7983  df-txp 36344  df-pprod 36345  df-cart 36355

This theorem is referenced by:  brimg  36427  brrestrict  36441