Theorem elxp6 3093

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 2640.

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 2640	. 2 ⊢ (A ∈ (B × C) ↔ (A = ⟨∪dom {A}, ∪ran {A}⟩ ∧ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C)))
2		1stval 3089	. . . . 5 ⊢ (1st ‘A) = ∪dom {A}
3		2ndval 3090	. . . . 5 ⊢ (2nd ‘A) = ∪ran {A}
4		opeq12 1878	. . . . 5 ⊢ (((1st ‘A) = ∪dom {A} ∧ (2nd ‘A) = ∪ran {A}) → ⟨(1st ‘A), (2nd ‘A)⟩ = ⟨∪dom {A}, ∪ran {A}⟩)
5	2, 3, 4	mp2an 520	. . . 4 ⊢ ⟨(1st ‘A), (2nd ‘A)⟩ = ⟨∪dom {A}, ∪ran {A}⟩
6	5	cleq2i 1111	. . 3 ⊢ (A = ⟨(1st ‘A), (2nd ‘A)⟩ ↔ A = ⟨∪dom {A}, ∪ran {A}⟩)
7	2	eleq1i 1152	. . . 4 ⊢ ((1st ‘A) ∈ B ↔ ∪dom {A} ∈ B)
8	3	eleq1i 1152	. . . 4 ⊢ ((2nd ‘A) ∈ C ↔ ∪ran {A} ∈ C)
9	7, 8	anbi12i 369	. . 3 ⊢ (((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ C) ↔ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C))
10	6, 9	anbi12i 369	. 2 ⊢ ((A = ⟨(1st ‘A), (2nd ‘A)⟩ ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ C)) ↔ (A = ⟨∪dom {A}, ∪ran {A}⟩ ∧ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C)))
11	1, 10	bitr4 154	1 ⊢ (A ∈ (B × C) ↔ (A = ⟨(1st ‘A), (2nd ‘A)⟩ ∧ ((1st ‘A) ∈ B ∧ (2nd ‘A) ∈ C)))

Syntax hints:   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  {csn 1808  ⟨cop 1810  ∪cuni 1919   × cxp 2408  dom cdm 2410  ran crn 2411   ‘cfv 2422  1^st c1st 3085  2^nd c2nd 3086

This theorem is referenced by:  ruclem13 4897  ruclem23 4907

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-1st 3087  df-2nd 3088

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