Theorem elxp6 7244

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7152. (Contributed by NM, 9-Oct-2004.)

Assertion

Ref	Expression
elxp6	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))

Proof of Theorem elxp6

Step	Hyp	Ref	Expression
1		elxp4 7152	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
2		1stval 7212	. . . . 5 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}
3		2ndval 7213	. . . . 5 ⊢ (2nd ‘𝐴) = ∪ ran {𝐴}
4	2, 3	opeq12i 4438	. . . 4 ⊢ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩
5	4	eqeq2i 2663	. . 3 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ↔ 𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩)
6	2	eleq1i 2721	. . . 4 ⊢ ((1st ‘𝐴) ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵)
7	3	eleq1i 2721	. . . 4 ⊢ ((2nd ‘𝐴) ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶)
8	6, 7	anbi12i 733	. . 3 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))
9	5, 8	anbi12i 733	. 2 ⊢ ((𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
10	1, 9	bitr4i 267	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))

Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {csn 4210  ⟨cop 4216  ∪ cuni 4468   × cxp 5141  dom cdm 5143  ran crn 5144  ‘cfv 5926  1^st c1st 7208  2^nd c2nd 7209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-1st 7210  df-2nd 7211

This theorem is referenced by:  elxp7  7245  eqopi  7246  1st2nd2  7249  r0weon  8873  qredeu  15419  qnumdencl  15494  setsstruct2  15943  tx1cn  21460  tx2cn  21461  txhaus  21498  psmetxrge0  22165  xppreima  29577  ofpreima2  29594  smatrcl  29990  1stmbfm  30450  2ndmbfm  30451  oddpwdcv  30545