Theorem elxp5 5109

Description: Membership in a cross product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 5108 when the double intersection does not create class existence problems (caused by int0 3854). (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 2746	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2		elex 2746	. . . 4 ⊢ (∩ ∩ 𝐴 ∈ 𝐵 → ∩ ∩ 𝐴 ∈ V)
3		elex 2746	. . . 4 ⊢ (∪ ran {𝐴} ∈ 𝐶 → ∪ ran {𝐴} ∈ V)
4	2, 3	anim12i 338	. . 3 ⊢ ((∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶) → (∩ ∩ 𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V))
5		opexg 4222	. . . . 5 ⊢ ((∩ ∩ 𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V) → ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∈ V)
6	5	adantl 277	. . . 4 ⊢ ((𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V)) → ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∈ V)
7		eleq1 2238	. . . . 5 ⊢ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ → (𝐴 ∈ V ↔ ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∈ V))
8	7	adantr 276	. . . 4 ⊢ ((𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V)) → (𝐴 ∈ V ↔ ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∈ V))
9	6, 8	mpbird 167	. . 3 ⊢ ((𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ V ∧ ∪ ran {𝐴} ∈ V)) → 𝐴 ∈ V)
10	4, 9	sylan2 286	. 2 ⊢ ((𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) → 𝐴 ∈ V)
11		elxp 4637	. . . 4 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
12		sneq 3600	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → {𝐴} = {⟨𝑥, 𝑦⟩})
13	12	rneqd 4849	. . . . . . . . . . . . 13 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ran {𝐴} = ran {⟨𝑥, 𝑦⟩})
14	13	unieqd 3816	. . . . . . . . . . . 12 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → ∪ ran {𝐴} = ∪ ran {⟨𝑥, 𝑦⟩})
15		vex 2738	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
16		vex 2738	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
17	15, 16	op2nda 5105	. . . . . . . . . . . 12 ⊢ ∪ ran {⟨𝑥, 𝑦⟩} = 𝑦
18	14, 17	eqtr2di 2225	. . . . . . . . . . 11 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ → 𝑦 = ∪ ran {𝐴})
19	18	pm4.71ri 392	. . . . . . . . . 10 ⊢ (𝐴 = ⟨𝑥, 𝑦⟩ ↔ (𝑦 = ∪ ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩))
20	19	anbi1i 458	. . . . . . . . 9 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑦 = ∪ ran {𝐴} ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
21		anass 401	. . . . . . . . 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 184	. . . . . . . 8 ⊢ ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
23	22	exbii 1603	. . . . . . 7 ⊢ (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑦(𝑦 = ∪ ran {𝐴} ∧ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))))
24		snexg 4179	. . . . . . . . . 10 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
25		rnexg 4885	. . . . . . . . . 10 ⊢ ({𝐴} ∈ V → ran {𝐴} ∈ V)
26	24, 25	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ V → ran {𝐴} ∈ V)
27		uniexg 4433	. . . . . . . . 9 ⊢ (ran {𝐴} ∈ V → ∪ ran {𝐴} ∈ V)
28	26, 27	syl 14	. . . . . . . 8 ⊢ (𝐴 ∈ V → ∪ ran {𝐴} ∈ V)
29		opeq2 3775	. . . . . . . . . . 11 ⊢ (𝑦 = ∪ ran {𝐴} → ⟨𝑥, 𝑦⟩ = ⟨𝑥, ∪ ran {𝐴}⟩)
30	29	eqeq2d 2187	. . . . . . . . . 10 ⊢ (𝑦 = ∪ ran {𝐴} → (𝐴 = ⟨𝑥, 𝑦⟩ ↔ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩))
31		eleq1 2238	. . . . . . . . . . 11 ⊢ (𝑦 = ∪ ran {𝐴} → (𝑦 ∈ 𝐶 ↔ ∪ ran {𝐴} ∈ 𝐶))
32	31	anbi2d 464	. . . . . . . . . 10 ⊢ (𝑦 = ∪ ran {𝐴} → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
33	30, 32	anbi12d 473	. . . . . . . . 9 ⊢ (𝑦 = ∪ ran {𝐴} → ((𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
34	33	ceqsexgv 2864	. . . . . . . 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	bitrid 192	. . . . . 6 ⊢ (𝐴 ∈ V → (∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
37		inteq 3843	. . . . . . . . . . . 12 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → ∩ 𝐴 = ∩ ⟨𝑥, ∪ ran {𝐴}⟩)
38	37	inteqd 3845	. . . . . . . . . . 11 ⊢ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, ∪ ran {𝐴}⟩)
39	38	adantl 277	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) → ∩ ∩ 𝐴 = ∩ ∩ ⟨𝑥, ∪ ran {𝐴}⟩)
40		op1stbg 4473	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ V ∧ ∪ ran {𝐴} ∈ V) → ∩ ∩ ⟨𝑥, ∪ ran {𝐴}⟩ = 𝑥)
41	15, 28, 40	sylancr 414	. . . . . . . . . . 11 ⊢ (𝐴 ∈ V → ∩ ∩ ⟨𝑥, ∪ ran {𝐴}⟩ = 𝑥)
42	41	adantr 276	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) → ∩ ∩ ⟨𝑥, ∪ ran {𝐴}⟩ = 𝑥)
43	39, 42	eqtr2d 2209	. . . . . . . . 9 ⊢ ((𝐴 ∈ V ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩) → 𝑥 = ∩ ∩ 𝐴)
44	43	ex 115	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ → 𝑥 = ∩ ∩ 𝐴))
45	44	pm4.71rd 394	. . . . . . 7 ⊢ (𝐴 ∈ V → (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ↔ (𝑥 = ∩ ∩ 𝐴 ∧ 𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩)))
46	45	anbi1d 465	. . . . . 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 401	. . . . . . 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 214	. . . . 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 1823	. . . 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	bitrid 192	. . 3 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥(𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))))
52		eqvisset 2745	. . . . . 6 ⊢ (𝑥 = ∩ ∩ 𝐴 → ∩ ∩ 𝐴 ∈ V)
53	52	adantr 276	. . . . 5 ⊢ ((𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) → ∩ ∩ 𝐴 ∈ V)
54	53	exlimiv 1596	. . . 4 ⊢ (∃𝑥(𝑥 = ∩ ∩ 𝐴 ∧ (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))) → ∩ ∩ 𝐴 ∈ V)
55	2	ad2antrl 490	. . . 4 ⊢ ((𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) → ∩ ∩ 𝐴 ∈ V)
56		opeq1 3774	. . . . . . 7 ⊢ (𝑥 = ∩ ∩ 𝐴 → ⟨𝑥, ∪ ran {𝐴}⟩ = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩)
57	56	eqeq2d 2187	. . . . . 6 ⊢ (𝑥 = ∩ ∩ 𝐴 → (𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ↔ 𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩))
58		eleq1 2238	. . . . . . 7 ⊢ (𝑥 = ∩ ∩ 𝐴 → (𝑥 ∈ 𝐵 ↔ ∩ ∩ 𝐴 ∈ 𝐵))
59	58	anbi1d 465	. . . . . 6 ⊢ (𝑥 = ∩ ∩ 𝐴 → ((𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶) ↔ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))
60	57, 59	anbi12d 473	. . . . 5 ⊢ (𝑥 = ∩ ∩ 𝐴 → ((𝐴 = ⟨𝑥, ∪ ran {𝐴}⟩ ∧ (𝑥 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
61	60	ceqsexgv 2864	. . . 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 704	. . 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 196	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶))))
64	1, 10, 63	pm5.21nii 704	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨∩ ∩ 𝐴, ∪ ran {𝐴}⟩ ∧ (∩ ∩ 𝐴 ∈ 𝐵 ∧ ∪ ran {𝐴} ∈ 𝐶)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1353  ∃wex 1490   ∈ wcel 2146  Vcvv 2735  {csn 3589  ⟨cop 3592  ∪ cuni 3805  ∩ cint 3840   × cxp 4618  ran crn 4621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630  df-rn 4631

This theorem is referenced by: (None)