Theorem elxp6 6331

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5224. (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		elex 2814	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2		opexg 4320	. . . 4 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ V)
3	2	adantl 277	. . 3 ⊢ ((𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ V)
4		eleq1 2294	. . . 4 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → (𝐴 ∈ V ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ V))
5	4	adantr 276	. . 3 ⊢ ((𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → (𝐴 ∈ V ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ V))
6	3, 5	mpbird 167	. 2 ⊢ ((𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
7		elxp4 5224	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
8		1stvalg 6304	. . . . . 6 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
9		2ndvalg 6305	. . . . . 6 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
10	8, 9	opeq12d 3870	. . . . 5 ⊢ (𝐴 ∈ V → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩)
11	10	eqeq2d 2243	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ↔ 𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩))
12	8	eleq1d 2300	. . . . 5 ⊢ (𝐴 ∈ V → ((1st ‘𝐴) ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵))
13	9	eleq1d 2300	. . . . 5 ⊢ (𝐴 ∈ V → ((2nd ‘𝐴) ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶))
14	12, 13	anbi12d 473	. . . 4 ⊢ (𝐴 ∈ V → (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
15	11, 14	anbi12d 473	. . 3 ⊢ (𝐴 ∈ V → ((𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
16	7, 15	bitr4id 199	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))))
17	1, 6, 16	pm5.21nii 711	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1397   ∈ wcel 2202  Vcvv 2802  {csn 3669  ⟨cop 3672  ∪ cuni 3893   × cxp 4723  dom cdm 4725  ran crn 4726  ‘cfv 5326  1^st c1st 6300  2^nd c2nd 6301

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fv 5334  df-1st 6302  df-2nd 6303

This theorem is referenced by:  elxp7  6332  oprssdmm  6333  eqopi  6334  1st2nd2  6337  eldju2ndl  7270  eldju2ndr  7271  aptap  8829  qredeu  12668  qnumdencl  12758  tx1cn  14992  tx2cn  14993  psmetxrge0  15055  xmetxpbl  15231