Theorem elxp4 2640

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp5 2641 and elxp6 3093.

Assertion

Ref	Expression
elxp4	⊢ (A ∈ (B × C) ↔ (A = ⟨∪dom {A}, ∪ran {A}⟩ ∧ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C)))

Proof of Theorem elxp4

Step	Hyp	Ref	Expression
1		elxp 2442	. 2 ⊢ (A ∈ (B × C) ↔ ∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)))
2		sneq 1816	. . . . . . . . . . . 12 ⊢ (A = ⟨x, y⟩ → {A} = {⟨x, y⟩})
3	2	rneqd 2557	. . . . . . . . . . 11 ⊢ (A = ⟨x, y⟩ → ran {A} = ran {⟨x, y⟩})
4	3	unieqd 1929	. . . . . . . . . 10 ⊢ (A = ⟨x, y⟩ → ∪ran {A} = ∪ran {⟨x, y⟩})
5		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
6		visset 1350	. . . . . . . . . . 11 ⊢ y ∈ V
7	5, 6	op2nda 2639	. . . . . . . . . 10 ⊢ ∪ran {⟨x, y⟩} = y
8	4, 7	syl6req 1141	. . . . . . . . 9 ⊢ (A = ⟨x, y⟩ → y = ∪ran {A})
9	8	pm4.71ri 484	. . . . . . . 8 ⊢ (A = ⟨x, y⟩ ↔ (y = ∪ran {A} ∧ A = ⟨x, y⟩))
10	9	anbi1i 368	. . . . . . 7 ⊢ ((A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ ((y = ∪ran {A} ∧ A = ⟨x, y⟩) ∧ (x ∈ B ∧ y ∈ C)))
11		anass 336	. . . . . . 7 ⊢ (((y = ∪ran {A} ∧ A = ⟨x, y⟩) ∧ (x ∈ B ∧ y ∈ C)) ↔ (y = ∪ran {A} ∧ (A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C))))
12	10, 11	bitr 151	. . . . . 6 ⊢ ((A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ (y = ∪ran {A} ∧ (A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C))))
13	12	biex 733	. . . . 5 ⊢ (∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ ∃y(y = ∪ran {A} ∧ (A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C))))
14		snex 1859	. . . . . . . 8 ⊢ {A} ∈ V
15		rnexg 2569	. . . . . . . 8 ⊢ ({A} ∈ V → ran {A} ∈ V)
16	14, 15	ax-mp 6	. . . . . . 7 ⊢ ran {A} ∈ V
17	16	uniex 1947	. . . . . 6 ⊢ ∪ran {A} ∈ V
18		opeq2 1877	. . . . . . . 8 ⊢ (y = ∪ran {A} → ⟨x, y⟩ = ⟨x, ∪ran {A}⟩)
19	18	cleq2d 1112	. . . . . . 7 ⊢ (y = ∪ran {A} → (A = ⟨x, y⟩ ↔ A = ⟨x, ∪ran {A}⟩))
20		eleq1 1149	. . . . . . . 8 ⊢ (y = ∪ran {A} → (y ∈ C ↔ ∪ran {A} ∈ C))
21	20	anbi2d 468	. . . . . . 7 ⊢ (y = ∪ran {A} → ((x ∈ B ∧ y ∈ C) ↔ (x ∈ B ∧ ∪ran {A} ∈ C)))
22	19, 21	anbi12d 476	. . . . . 6 ⊢ (y = ∪ran {A} → ((A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C))))
23	17, 22	ceqsexv 1371	. . . . 5 ⊢ (∃y(y = ∪ran {A} ∧ (A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C))) ↔ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C)))
24	13, 23	bitr 151	. . . 4 ⊢ (∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C)))
25		sneq 1816	. . . . . . . . 9 ⊢ (A = ⟨x, ∪ran {A}⟩ → {A} = {⟨x, ∪ran {A}⟩})
26	25	dmeqd 2533	. . . . . . . 8 ⊢ (A = ⟨x, ∪ran {A}⟩ → dom {A} = dom {⟨x, ∪ran {A}⟩})
27	26	unieqd 1929	. . . . . . 7 ⊢ (A = ⟨x, ∪ran {A}⟩ → ∪dom {A} = ∪dom {⟨x, ∪ran {A}⟩})
28	5	op1sta 2635	. . . . . . 7 ⊢ ∪dom {⟨x, ∪ran {A}⟩} = x
29	27, 28	syl6req 1141	. . . . . 6 ⊢ (A = ⟨x, ∪ran {A}⟩ → x = ∪dom {A})
30	29	pm4.71ri 484	. . . . 5 ⊢ (A = ⟨x, ∪ran {A}⟩ ↔ (x = ∪dom {A} ∧ A = ⟨x, ∪ran {A}⟩))
31	30	anbi1i 368	. . . 4 ⊢ ((A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C)) ↔ ((x = ∪dom {A} ∧ A = ⟨x, ∪ran {A}⟩) ∧ (x ∈ B ∧ ∪ran {A} ∈ C)))
32		anass 336	. . . 4 ⊢ (((x = ∪dom {A} ∧ A = ⟨x, ∪ran {A}⟩) ∧ (x ∈ B ∧ ∪ran {A} ∈ C)) ↔ (x = ∪dom {A} ∧ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C))))
33	24, 31, 32	3bitr 155	. . 3 ⊢ (∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ (x = ∪dom {A} ∧ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C))))
34	33	biex 733	. 2 ⊢ (∃x∃y(A = ⟨x, y⟩ ∧ (x ∈ B ∧ y ∈ C)) ↔ ∃x(x = ∪dom {A} ∧ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C))))
35		dmexg 2551	. . . . 5 ⊢ ({A} ∈ V → dom {A} ∈ V)
36	14, 35	ax-mp 6	. . . 4 ⊢ dom {A} ∈ V
37	36	uniex 1947	. . 3 ⊢ ∪dom {A} ∈ V
38		opeq1 1876	. . . . 5 ⊢ (x = ∪dom {A} → ⟨x, ∪ran {A}⟩ = ⟨∪dom {A}, ∪ran {A}⟩)
39	38	cleq2d 1112	. . . 4 ⊢ (x = ∪dom {A} → (A = ⟨x, ∪ran {A}⟩ ↔ A = ⟨∪dom {A}, ∪ran {A}⟩))
40		eleq1 1149	. . . . 5 ⊢ (x = ∪dom {A} → (x ∈ B ↔ ∪dom {A} ∈ B))
41	40	anbi1d 469	. . . 4 ⊢ (x = ∪dom {A} → ((x ∈ B ∧ ∪ran {A} ∈ C) ↔ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C)))
42	39, 41	anbi12d 476	. . 3 ⊢ (x = ∪dom {A} → ((A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C)) ↔ (A = ⟨∪dom {A}, ∪ran {A}⟩ ∧ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C))))
43	37, 42	ceqsexv 1371	. 2 ⊢ (∃x(x = ∪dom {A} ∧ (A = ⟨x, ∪ran {A}⟩ ∧ (x ∈ B ∧ ∪ran {A} ∈ C))) ↔ (A = ⟨∪dom {A}, ∪ran {A}⟩ ∧ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C)))
44	1, 34, 43	3bitr 155	1 ⊢ (A ∈ (B × C) ↔ (A = ⟨∪dom {A}, ∪ran {A}⟩ ∧ (∪dom {A} ∈ B ∧ ∪ran {A} ∈ C)))

Syntax hints:   ↔ wb 127   ∧ wa 196  ∃wex 678   = wceq 1091   ∈ wcel 1092  Vcvv 1348  {csn 1808  ⟨cop 1810  ∪cuni 1919   × cxp 2408  dom cdm 2410  ran crn 2411

This theorem is referenced by:  elxp6 3093  xpdom2 3345  xpmapenlem3 3393  xpmapenlem5 3395

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-clab 1093  df-cleq 1097  df-clel 1099  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-xp 2424  df-rel 2425  df-cnv 2426  df-dm 2428  df-rn 2429

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