Theorem elxp5 5092

Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5091 when the double intersection does not create class existence problems (caused by int0 3838). (Contributed by NM, 1-Aug-2004.)

Assertion

Ref	Expression
elxp5	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))

Proof of Theorem elxp5

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

Step	Hyp	Ref	Expression
1		elex 2737	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2		elex 2737	. . . 4 ⊢ (∩ ∩ 𝐴 ∈ 𝐵 → ∩ ∩ 𝐴 ∈ V)
3		elex 2737	. . . 4 ⊢ (∪ ran {𝐴} ∈ 𝐶 → ∪ ran {𝐴} ∈ V)
4	2, 3	anim12i 336	. . 3 ⊢ ((∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶) → (∩ ∩ 𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V))
5		opexg 4206	. . . . 5 ⊢ ((∩ ∩ 𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V) → ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∈ V)
6	5	adantl 275	. . . 4 ⊢ ((𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V)) → ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∈ V)
7		eleq1 2229	. . . . 5 ⊢ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ → (𝐴 ∈ V ↔ ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∈ V))
8	7	adantr 274	. . . 4 ⊢ ((𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V)) → (𝐴 ∈ V ↔ ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∈ V))
9	6, 8	mpbird 166	. . 3 ⊢ ((𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V)) → 𝐴 ∈ V)
10	4, 9	sylan2 284	. 2 ⊢ ((𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) → 𝐴 ∈ V)
11		elxp 4621	. . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
12		sneq 3587	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
13	12	rneqd 4833	. . . . . . . . . . . . 13 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
14	13	unieqd 3800	. . . . . . . . . . . 12 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ ran {𝐴} = ∪ ran {⟨𝑥, 𝑦⟩})
15		vex 2729	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
16		vex 2729	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
17	15, 16	op2nda 5088	. . . . . . . . . . . 12 ⊢ ∪ ran {⟨𝑥, 𝑦⟩} = 𝑦
18	14, 17	eqtr2di 2216	. . . . . . . . . . 11 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = ∪ ran {𝐴})
19	18	pm4.71ri 390	. . . . . . . . . 10 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = ∪ ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
20	19	anbi1i 454	. . . . . . . . 9 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑦 = ∪ ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
21		anass 399	. . . . . . . . 9 ⊢ (((𝑦 = ∪ ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
22	20, 21	bitri 183	. . . . . . . 8 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
23	22	exbii 1593	. . . . . . 7 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
24		snexg 4163	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
25		rnexg 4869	. . . . . . . . . 10 ⊢ ({𝐴} ∈ V → ran {𝐴} ∈ V)
26	24, 25	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ran {𝐴} ∈ V)
27		uniexg 4417	. . . . . . . . 9 ⊢ (ran {𝐴} ∈ V → ∪ ran {𝐴} ∈ V)
28	26, 27	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∪ ran {𝐴} ∈ V)
29		opeq2 3759	. . . . . . . . . . 11 ⊢ (𝑦 = ∪ ran {𝐴} → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∪ ran {𝐴}⟩)
30	29	eqeq2d 2177	. . . . . . . . . 10 ⊢ (𝑦 = ∪ ran {𝐴} → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩))
31		eleq1 2229	. . . . . . . . . . 11 ⊢ (𝑦 = ∪ ran {𝐴} → (𝑦 ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶))
32	31	anbi2d 460	. . . . . . . . . 10 ⊢ (𝑦 = ∪ ran {𝐴} → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
33	30, 32	anbi12d 465	. . . . . . . . 9 ⊢ (𝑦 = ∪ ran {𝐴} → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
34	33	ceqsexgv 2855	. . . . . . . 8 ⊢ (∪ ran {𝐴} ∈ V → (∃𝑦(𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
35	28, 34	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ V → (∃𝑦(𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
36	23, 35	syl5bb 191	. . . . . 6 ⊢ (𝐴 ∈ V → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
37		inteq 3827	. . . . . . . . . . . 12 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → ∩ 𝐴 = ∩ ⟨𝑥, ∪ ran {𝐴}⟩)
38	37	inteqd 3829	. . . . . . . . . . 11 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, ∪ ran {𝐴}⟩)
39	38	adantl 275	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, ∪ ran {𝐴}⟩)
40		op1stbg 4457	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ ∪ ran {𝐴} ∈ V) → ∩ ∩ ⟨𝑥, ∪ ran {𝐴}⟩ = 𝑥)
41	15, 28, 40	sylancr 411	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → ∩ ∩ ⟨𝑥, ∪ ran {𝐴}⟩ = 𝑥)
42	41	adantr 274	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) → ∩ ∩ ⟨𝑥, ∪ ran {𝐴}⟩ = 𝑥)
43	39, 42	eqtr2d 2199	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) → 𝑥 = ∩ ∩ 𝐴)
44	43	ex 114	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → 𝑥 = ∩ ∩ 𝐴))
45	44	pm4.71rd 392	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ↔ (𝑥 = ∩ ∩ 𝐴 ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩)))
46	45	anbi1d 461	. . . . . 6 ⊢ (𝐴 ∈ V → ((𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) ↔ ((𝑥 = ∩ ∩ 𝐴 ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
47		anass 399	. . . . . . 7 ⊢ (((𝑥 = ∩ ∩ 𝐴 ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) ↔ (𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
48	47	a1i 9	. . . . . 6 ⊢ (𝐴 ∈ V → (((𝑥 = ∩ ∩ 𝐴 ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) ↔ (𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))))
49	36, 46, 48	3bitrd 213	. . . . 5 ⊢ (𝐴 ∈ V → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))))
50	49	exbidv 1813	. . . 4 ⊢ (𝐴 ∈ V → (∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑥(𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))))
51	11, 50	syl5bb 191	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥(𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))))
52		eqvisset 2736	. . . . . 6 ⊢ (𝑥 = ∩ ∩ 𝐴 → ∩ ∩ 𝐴 ∈ V)
53	52	adantr 274	. . . . 5 ⊢ ((𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) → ∩ ∩ 𝐴 ∈ V)
54	53	exlimiv 1586	. . . 4 ⊢ (∃𝑥(𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) → ∩ ∩ 𝐴 ∈ V)
55	2	ad2antrl 482	. . . 4 ⊢ ((𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) → ∩ ∩ 𝐴 ∈ V)
56		opeq1 3758	. . . . . . 7 ⊢ (𝑥 = ∩ ∩ 𝐴 → ⟨𝑥, ∪ ran {𝐴}⟩ = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩)
57	56	eqeq2d 2177	. . . . . 6 ⊢ (𝑥 = ∩ ∩ 𝐴 → (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ↔ 𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩))
58		eleq1 2229	. . . . . . 7 ⊢ (𝑥 = ∩ ∩ 𝐴 → (𝑥 ∈ 𝐵 ↔ ∩ ∩ 𝐴 ∈ 𝐵))
59	58	anbi1d 461	. . . . . 6 ⊢ (𝑥 = ∩ ∩ 𝐴 → ((𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶) ↔ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
60	57, 59	anbi12d 465	. . . . 5 ⊢ (𝑥 = ∩ ∩ 𝐴 → ((𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
61	60	ceqsexgv 2855	. . . 4 ⊢ (∩ ∩ 𝐴 ∈ V → (∃𝑥(𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
62	54, 55, 61	pm5.21nii 694	. . 3 ⊢ (∃𝑥(𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
63	51, 62	bitrdi 195	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
64	1, 10, 63	pm5.21nii 694	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726  {csn 3576  ⟨cop 3579  ∪ cuni 3789  ∩ cint 3824   × cxp 4602  ran crn 4605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by: (None)