Theorem elxp4 7678

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7679, elxp6 7773, and elxp7 7774. (Contributed by NM, 17-Feb-2004.)

Assertion

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

Proof of Theorem elxp4

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 5559	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
2		sneq 4537	. . . . . . . . . . . 12 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
3	2	rneqd 5792	. . . . . . . . . . 11 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
4	3	unieqd 4819	. . . . . . . . . 10 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ ran {𝐴} = ∪ ran {⟨𝑥, 𝑦⟩})
5		vex 3402	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
6		vex 3402	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7	5, 6	op2nda 6071	. . . . . . . . . 10 ⊢ ∪ ran {⟨𝑥, 𝑦⟩} = 𝑦
8	4, 7	eqtr2di 2788	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = ∪ ran {𝐴})
9	8	pm4.71ri 564	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = ∪ ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
10	9	anbi1i 627	. . . . . . 7 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑦 = ∪ ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
11		anass 472	. . . . . . 7 ⊢ (((𝑦 = ∪ ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
12	10, 11	bitri 278	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
13	12	exbii 1855	. . . . 5 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
14		snex 5309	. . . . . . . 8 ⊢ {𝐴} ∈ V
15	14	rnex 7668	. . . . . . 7 ⊢ ran {𝐴} ∈ V
16	15	uniex 7507	. . . . . 6 ⊢ ∪ ran {𝐴} ∈ V
17		opeq2 4771	. . . . . . . 8 ⊢ (𝑦 = ∪ ran {𝐴} → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∪ ran {𝐴}⟩)
18	17	eqeq2d 2747	. . . . . . 7 ⊢ (𝑦 = ∪ ran {𝐴} → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩))
19		eleq1 2818	. . . . . . . 8 ⊢ (𝑦 = ∪ ran {𝐴} → (𝑦 ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶))
20	19	anbi2d 632	. . . . . . 7 ⊢ (𝑦 = ∪ ran {𝐴} → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
21	18, 20	anbi12d 634	. . . . . 6 ⊢ (𝑦 = ∪ ran {𝐴} → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
22	16, 21	ceqsexv 3445	. . . . 5 ⊢ (∃𝑦(𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
23	13, 22	bitri 278	. . . 4 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
24		sneq 4537	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → {𝐴} = {⟨𝑥, ∪ ran {𝐴}⟩})
25	24	dmeqd 5759	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → dom {𝐴} = dom {⟨𝑥, ∪ ran {𝐴}⟩})
26	25	unieqd 4819	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → ∪ dom {𝐴} = ∪ dom {⟨𝑥, ∪ ran {𝐴}⟩})
27	5, 16	op1sta 6068	. . . . . . 7 ⊢ ∪ dom {⟨𝑥, ∪ ran {𝐴}⟩} = 𝑥
28	26, 27	eqtr2di 2788	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → 𝑥 = ∪ dom {𝐴})
29	28	pm4.71ri 564	. . . . 5 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ↔ (𝑥 = ∪ dom {𝐴} ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩))
30	29	anbi1i 627	. . . 4 ⊢ ((𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) ↔ ((𝑥 = ∪ dom {𝐴} ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
31		anass 472	. . . 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))))
32	23, 30, 31	3bitri 300	. . 3 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 = ∪ dom {𝐴} ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
33	32	exbii 1855	. 2 ⊢ (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑥(𝑥 = ∪ dom {𝐴} ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
34	14	dmex 7667	. . . 4 ⊢ dom {𝐴} ∈ V
35	34	uniex 7507	. . 3 ⊢ ∪ dom {𝐴} ∈ V
36		opeq1 4770	. . . . 5 ⊢ (𝑥 = ∪ dom {𝐴} → ⟨𝑥, ∪ ran {𝐴}⟩ = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩)
37	36	eqeq2d 2747	. . . 4 ⊢ (𝑥 = ∪ dom {𝐴} → (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ↔ 𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩))
38		eleq1 2818	. . . . 5 ⊢ (𝑥 = ∪ dom {𝐴} → (𝑥 ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵))
39	38	anbi1d 633	. . . 4 ⊢ (𝑥 = ∪ dom {𝐴} → ((𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
40	37, 39	anbi12d 634	. . 3 ⊢ (𝑥 = ∪ dom {𝐴} → ((𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
41	35, 40	ceqsexv 3445	. 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)))
42	1, 33, 41	3bitri 300	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1543  ∃wex 1787   ∈ wcel 2112  {csn 4527  ⟨cop 4533  ∪ cuni 4805   × cxp 5534  dom cdm 5536  ran crn 5537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-br 5040  df-opab 5102  df-xp 5542  df-rel 5543  df-cnv 5544  df-dm 5546  df-rn 5547

This theorem is referenced by:  elxp6  7773  xpdom2  8718