Theorem elxp4 7930

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp5 7931, elxp6 8028, and elxp7 8029. (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 5701	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
2		sneq 4640	. . . . . . . . . . . 12 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
3	2	rneqd 5940	. . . . . . . . . . 11 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
4	3	unieqd 4922	. . . . . . . . . 10 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ ran {𝐴} = ∪ ran {⟨𝑥, 𝑦⟩})
5		vex 3465	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
6		vex 3465	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
7	5, 6	op2nda 6234	. . . . . . . . . 10 ⊢ ∪ ran {⟨𝑥, 𝑦⟩} = 𝑦
8	4, 7	eqtr2di 2782	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = ∪ ran {𝐴})
9	8	pm4.71ri 559	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = ∪ ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
10	9	anbi1i 622	. . . . . . 7 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑦 = ∪ ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
11		anass 467	. . . . . . 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 274	. . . . . 6 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
13	12	exbii 1842	. . . . 5 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
14		snex 5433	. . . . . . . 8 ⊢ {𝐴} ∈ V
15	14	rnex 7918	. . . . . . 7 ⊢ ran {𝐴} ∈ V
16	15	uniex 7747	. . . . . 6 ⊢ ∪ ran {𝐴} ∈ V
17		opeq2 4876	. . . . . . . 8 ⊢ (𝑦 = ∪ ran {𝐴} → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∪ ran {𝐴}⟩)
18	17	eqeq2d 2736	. . . . . . 7 ⊢ (𝑦 = ∪ ran {𝐴} → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩))
19		eleq1 2813	. . . . . . . 8 ⊢ (𝑦 = ∪ ran {𝐴} → (𝑦 ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶))
20	19	anbi2d 628	. . . . . . 7 ⊢ (𝑦 = ∪ ran {𝐴} → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
21	18, 20	anbi12d 630	. . . . . 6 ⊢ (𝑦 = ∪ ran {𝐴} → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
22	16, 21	ceqsexv 3514	. . . . 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 274	. . . 4 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
24		sneq 4640	. . . . . . . . 9 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → {𝐴} = {⟨𝑥, ∪ ran {𝐴}⟩})
25	24	dmeqd 5908	. . . . . . . 8 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → dom {𝐴} = dom {⟨𝑥, ∪ ran {𝐴}⟩})
26	25	unieqd 4922	. . . . . . 7 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → ∪ dom {𝐴} = ∪ dom {⟨𝑥, ∪ ran {𝐴}⟩})
27	5, 16	op1sta 6231	. . . . . . 7 ⊢ ∪ dom {⟨𝑥, ∪ ran {𝐴}⟩} = 𝑥
28	26, 27	eqtr2di 2782	. . . . . 6 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → 𝑥 = ∪ dom {𝐴})
29	28	pm4.71ri 559	. . . . 5 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ↔ (𝑥 = ∪ dom {𝐴} ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩))
30	29	anbi1i 622	. . . 4 ⊢ ((𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) ↔ ((𝑥 = ∪ dom {𝐴} ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
31		anass 467	. . . 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 296	. . 3 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 = ∪ dom {𝐴} ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
33	32	exbii 1842	. 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 7917	. . . 4 ⊢ dom {𝐴} ∈ V
35	34	uniex 7747	. . 3 ⊢ ∪ dom {𝐴} ∈ V
36		opeq1 4875	. . . . 5 ⊢ (𝑥 = ∪ dom {𝐴} → ⟨𝑥, ∪ ran {𝐴}⟩ = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩)
37	36	eqeq2d 2736	. . . 4 ⊢ (𝑥 = ∪ dom {𝐴} → (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ↔ 𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩))
38		eleq1 2813	. . . . 5 ⊢ (𝑥 = ∪ dom {𝐴} → (𝑥 ∈ 𝐵 ↔ ∪ dom {𝐴} ∈ 𝐵))
39	38	anbi1d 629	. . . 4 ⊢ (𝑥 = ∪ dom {𝐴} → ((𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶) ↔ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
40	37, 39	anbi12d 630	. . 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 3514	. 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 296	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∪ dom {𝐴}, ∪ ran {𝐴}⟩ ∧ (∪ dom {𝐴} ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {csn 4630  ⟨cop 4636  ∪ cuni 4909   × cxp 5676  dom cdm 5678  ran crn 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-cnv 5686  df-dm 5688  df-rn 5689

This theorem is referenced by:  elxp6  8028  xpdom2  9092