Theorem xp2 6226

Description: Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)

Assertion

Ref	Expression
xp2	⊢ (𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵)}

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem xp2

Step	Hyp	Ref	Expression
1		elxp7 6223	. . 3 ⊢ (𝑥 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵)))
2	1	abbi2i 2308	. 2 ⊢ (𝐴 × 𝐵) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵))}
3		df-rab 2481	. 2 ⊢ {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵))}
4	2, 3	eqtr4i 2217	1 ⊢ (𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st ‘𝑥) ∈ 𝐴 ∧ (2nd ‘𝑥) ∈ 𝐵)}

Syntax hints:   ∧ wa 104   = wceq 1364   ∈ wcel 2164  {cab 2179  {crab 2476  Vcvv 2760   × cxp 4657  ‘cfv 5254  1^st c1st 6191  2^nd c2nd 6192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fo 5260  df-fv 5262  df-1st 6193  df-2nd 6194

This theorem is referenced by:  unielxp  6227