Theorem elxp6 5796

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4808. (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 2566	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2		opexg 3964	. . . 4 ⊢ (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ V)
3	2	adantl 262	. . 3 ⊢ ((𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ V)
4		eleq1 2100	. . . 4 ⊢ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ → (𝐴 ∈ V ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ V))
5	4	adantr 261	. . 3 ⊢ ((𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → (𝐴 ∈ V ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∈ V))
6	3, 5	mpbird 156	. 2 ⊢ ((𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
7		1stvalg 5769	. . . . . 6 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
8		2ndvalg 5770	. . . . . 6 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) = ∪ ran {𝐴})
9	7, 8	opeq12d 3557	. . . . 5 ⊢ (𝐴 ∈ V → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩)
10	9	eqeq2d 2051	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ↔ 𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩))
11	7	eleq1d 2106	. . . . 5 ⊢ (𝐴 ∈ V → ((1st ‘𝐴) ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵))
12	8	eleq1d 2106	. . . . 5 ⊢ (𝐴 ∈ V → ((2nd ‘𝐴) ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶))
13	11, 12	anbi12d 442	. . . 4 ⊢ (𝐴 ∈ V → (((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
14	10, 13	anbi12d 442	. . 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))))
15		elxp4 4808	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
16	14, 15	syl6rbbr 188	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))))
17	1, 6, 16	pm5.21nii 620	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375  ⟨cop 3378  ∪ cuni 3580   × cxp 4343  dom cdm 4345  ran crn 4346  ‘cfv 4902  1^st c1st 5765  2^nd c2nd 5766

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-1st 5767  df-2nd 5768

This theorem is referenced by:  elxp7  5797  eqopi  5798  1st2nd2  5801