Theorem elxp7 6314

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5215. (Contributed by NM, 19-Aug-2006.)

Assertion

Ref	Expression
elxp7	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))

Proof of Theorem elxp7

Step	Hyp	Ref	Expression
1		elex 2811	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2		elex 2811	. . 3 ⊢ (𝐴 ∈ (V × V) → 𝐴 ∈ V)
3	2	adantr 276	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
4		elxp6 6313	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
5		elxp6 6313	. . . . 5 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V)))
6		1stexg 6311	. . . . . . 7 ⊢ (𝐴 ∈ V → (1st ‘𝐴) ∈ V)
7		2ndexg 6312	. . . . . . 7 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) ∈ V)
8	6, 7	jca 306	. . . . . 6 ⊢ (𝐴 ∈ V → ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V))
9	8	biantrud 304	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V))))
10	5, 9	bitr4id 199	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩))
11	10	anbi1d 465	. . 3 ⊢ (𝐴 ∈ V → ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))))
12	4, 11	bitr4id 199	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))))
13	1, 3, 12	pm5.21nii 709	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395   ∈ wcel 2200  Vcvv 2799  ⟨cop 3669   × cxp 4716  ‘cfv 5317  1^st c1st 6282  2^nd c2nd 6283

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-fo 5323  df-fv 5325  df-1st 6284  df-2nd 6285

This theorem is referenced by:  xp2  6317  unielxp  6318  1stconst  6365  2ndconst  6366  f1od2  6379