Theorem disjrnmpt2 45095

Description: Disjointness of the range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
disjrnmpt2.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
disjrnmpt2	⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran 𝐹 𝑦)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐹

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐹(𝑥)

Proof of Theorem disjrnmpt2

Dummy variables 𝑢 𝑧 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . . 6 ⊢ (𝑦 = 𝑤 → 𝑦 = 𝑤)
2	1	cbvdisjv 5144	. . . . 5 ⊢ (Disj 𝑦 ∈ ran 𝐹 𝑦 ↔ Disj 𝑤 ∈ ran 𝐹 𝑤)
3		id 22	. . . . . . 7 ⊢ (𝑤 = 𝑣 → 𝑤 = 𝑣)
4	3	ndisj2 44953	. . . . . 6 ⊢ (¬ Disj 𝑤 ∈ ran 𝐹 𝑤 ↔ ∃𝑤 ∈ ran 𝐹∃𝑣 ∈ ran 𝐹(𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅))
5	4	biimpi 216	. . . . 5 ⊢ (¬ Disj 𝑤 ∈ ran 𝐹 𝑤 → ∃𝑤 ∈ ran 𝐹∃𝑣 ∈ ran 𝐹(𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅))
6	2, 5	sylnbi 330	. . . 4 ⊢ (¬ Disj 𝑦 ∈ ran 𝐹 𝑦 → ∃𝑤 ∈ ran 𝐹∃𝑣 ∈ ran 𝐹(𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅))
7		disjrnmpt2.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
8	7	elrnmpt 5981	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ ran 𝐹 → (𝑤 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝑤 = 𝐵))
9	8	ibi 267	. . . . . . . . . . 11 ⊢ (𝑤 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
10		nfcv 2908	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧𝐵
11		nfcsb1v 3946	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑧 / 𝑥⦌𝐵
12		csbeq1a 3935	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
13	10, 11, 12	cbvmpt 5277	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝐵)
14	7, 13	eqtri 2768	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑧 ∈ 𝐴 ↦ ⦋𝑧 / 𝑥⦌𝐵)
15	14	elrnmpt 5981	. . . . . . . . . . . 12 ⊢ (𝑣 ∈ ran 𝐹 → (𝑣 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝐴 𝑣 = ⦋𝑧 / 𝑥⦌𝐵))
16	15	ibi 267	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ran 𝐹 → ∃𝑧 ∈ 𝐴 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)
17	9, 16	anim12i 612	. . . . . . . . . 10 ⊢ ((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) → (∃𝑥 ∈ 𝐴 𝑤 = 𝐵 ∧ ∃𝑧 ∈ 𝐴 𝑣 = ⦋𝑧 / 𝑥⦌𝐵))
18		nfv 1913	. . . . . . . . . . 11 ⊢ Ⅎ𝑧 𝑤 = 𝐵
19	11	nfeq2 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑣 = ⦋𝑧 / 𝑥⦌𝐵
20	18, 19	reean 3322	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) ↔ (∃𝑥 ∈ 𝐴 𝑤 = 𝐵 ∧ ∃𝑧 ∈ 𝐴 𝑣 = ⦋𝑧 / 𝑥⦌𝐵))
21	17, 20	sylibr 234	. . . . . . . . 9 ⊢ ((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵))
22	21	adantr 480	. . . . . . . 8 ⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵))
23		nfmpt1 5274	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵)
24	7, 23	nfcxfr 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝐹
25	24	nfrn 5977	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥ran 𝐹
26	25	nfcri 2900	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 ∈ ran 𝐹
27	25	nfcri 2900	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑣 ∈ ran 𝐹
28	26, 27	nfan 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹)
29		nfv 1913	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)
30	28, 29	nfan 1898	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅))
31		simpll 766	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑥 = 𝑧) → 𝑤 = 𝐵)
32	12	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑥 = 𝑧) → 𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
33		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑣 = ⦋𝑧 / 𝑥⦌𝐵 → 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)
34	33	eqcomd 2746	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑣 = ⦋𝑧 / 𝑥⦌𝐵 → ⦋𝑧 / 𝑥⦌𝐵 = 𝑣)
35	34	ad2antlr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑥 = 𝑧) → ⦋𝑧 / 𝑥⦌𝐵 = 𝑣)
36	31, 32, 35	3eqtrd 2784	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) ∧ 𝑥 = 𝑧) → 𝑤 = 𝑣)
37	36	adantll 713	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) ∧ 𝑥 = 𝑧) → 𝑤 = 𝑣)
38		simpll 766	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) ∧ 𝑥 = 𝑧) → 𝑤 ≠ 𝑣)
39	38	neneqd 2951	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) ∧ 𝑥 = 𝑧) → ¬ 𝑤 = 𝑣)
40	37, 39	pm2.65da 816	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ≠ 𝑣 ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → ¬ 𝑥 = 𝑧)
41	40	neqned 2953	. . . . . . . . . . . . . . 15 ⊢ ((𝑤 ≠ 𝑣 ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → 𝑥 ≠ 𝑧)
42	41	adantlr 714	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → 𝑥 ≠ 𝑧)
43		id 22	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝐵 → 𝑤 = 𝐵)
44	43	eqcomd 2746	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = 𝐵 → 𝐵 = 𝑤)
45	44	ad2antrl 727	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∩ 𝑣) ≠ ∅ ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → 𝐵 = 𝑤)
46	34	ad2antll 728	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑤 ∩ 𝑣) ≠ ∅ ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → ⦋𝑧 / 𝑥⦌𝐵 = 𝑣)
47	45, 46	ineq12d 4242	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∩ 𝑣) ≠ ∅ ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = (𝑤 ∩ 𝑣))
48		simpl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝑤 ∩ 𝑣) ≠ ∅ ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → (𝑤 ∩ 𝑣) ≠ ∅)
49	47, 48	eqnetrd 3014	. . . . . . . . . . . . . . 15 ⊢ (((𝑤 ∩ 𝑣) ≠ ∅ ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)
50	49	adantll 713	. . . . . . . . . . . . . 14 ⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)
51	42, 50	jca 511	. . . . . . . . . . . . 13 ⊢ (((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) ∧ (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵)) → (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))
52	51	ex 412	. . . . . . . . . . . 12 ⊢ ((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) → ((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) → (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)))
53	52	adantl 481	. . . . . . . . . . 11 ⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → ((𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) → (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)))
54	53	reximdv 3176	. . . . . . . . . 10 ⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → (∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) → ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)))
55	54	a1d 25	. . . . . . . . 9 ⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → (𝑥 ∈ 𝐴 → (∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) → ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))))
56	30, 55	reximdai 3267	. . . . . . . 8 ⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑤 = 𝐵 ∧ 𝑣 = ⦋𝑧 / 𝑥⦌𝐵) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)))
57	22, 56	mpd 15	. . . . . . 7 ⊢ (((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) ∧ (𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅)) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))
58	57	ex 412	. . . . . 6 ⊢ ((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) → ((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)))
59	58	a1i 11	. . . . 5 ⊢ (¬ Disj 𝑦 ∈ ran 𝐹 𝑦 → ((𝑤 ∈ ran 𝐹 ∧ 𝑣 ∈ ran 𝐹) → ((𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))))
60	59	rexlimdvv 3218	. . . 4 ⊢ (¬ Disj 𝑦 ∈ ran 𝐹 𝑦 → (∃𝑤 ∈ ran 𝐹∃𝑣 ∈ ran 𝐹(𝑤 ≠ 𝑣 ∧ (𝑤 ∩ 𝑣) ≠ ∅) → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)))
61	6, 60	mpd 15	. . 3 ⊢ (¬ Disj 𝑦 ∈ ran 𝐹 𝑦 → ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))
62		csbeq1 3924	. . . . . 6 ⊢ (𝑢 = 𝑧 → ⦋𝑢 / 𝑥⦌𝐵 = ⦋𝑧 / 𝑥⦌𝐵)
63	62	ndisj2 44953	. . . . 5 ⊢ (¬ Disj 𝑢 ∈ 𝐴 ⦋𝑢 / 𝑥⦌𝐵 ↔ ∃𝑢 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))
64		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑥𝐴
65		nfv 1913	. . . . . . . 8 ⊢ Ⅎ𝑥 𝑢 ≠ 𝑧
66		nfcsb1v 3946	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐵
67	66, 11	nfin 4245	. . . . . . . . 9 ⊢ Ⅎ𝑥(⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)
68		nfcv 2908	. . . . . . . . 9 ⊢ Ⅎ𝑥∅
69	67, 68	nfne 3049	. . . . . . . 8 ⊢ Ⅎ𝑥(⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅
70	65, 69	nfan 1898	. . . . . . 7 ⊢ Ⅎ𝑥(𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)
71	64, 70	nfrexw 3319	. . . . . 6 ⊢ Ⅎ𝑥∃𝑧 ∈ 𝐴 (𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)
72		nfv 1913	. . . . . 6 ⊢ Ⅎ𝑢∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)
73		neeq1 3009	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → (𝑢 ≠ 𝑧 ↔ 𝑥 ≠ 𝑧))
74		csbeq1 3924	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑥 → ⦋𝑢 / 𝑥⦌𝐵 = ⦋𝑥 / 𝑥⦌𝐵)
75		csbid 3934	. . . . . . . . . . 11 ⊢ ⦋𝑥 / 𝑥⦌𝐵 = 𝐵
76	74, 75	eqtrdi 2796	. . . . . . . . . 10 ⊢ (𝑢 = 𝑥 → ⦋𝑢 / 𝑥⦌𝐵 = 𝐵)
77	76	ineq1d 4240	. . . . . . . . 9 ⊢ (𝑢 = 𝑥 → (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵))
78	77	neeq1d 3006	. . . . . . . 8 ⊢ (𝑢 = 𝑥 → ((⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅ ↔ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))
79	73, 78	anbi12d 631	. . . . . . 7 ⊢ (𝑢 = 𝑥 → ((𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) ↔ (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)))
80	79	rexbidv 3185	. . . . . 6 ⊢ (𝑢 = 𝑥 → (∃𝑧 ∈ 𝐴 (𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) ↔ ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅)))
81	71, 72, 80	cbvrexw 3313	. . . . 5 ⊢ (∃𝑢 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑢 ≠ 𝑧 ∧ (⦋𝑢 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅) ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))
82	63, 81	bitri 275	. . . 4 ⊢ (¬ Disj 𝑢 ∈ 𝐴 ⦋𝑢 / 𝑥⦌𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))
83		nfcv 2908	. . . . 5 ⊢ Ⅎ𝑢𝐵
84		csbeq1a 3935	. . . . 5 ⊢ (𝑥 = 𝑢 → 𝐵 = ⦋𝑢 / 𝑥⦌𝐵)
85	83, 66, 84	cbvdisj 5143	. . . 4 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ Disj 𝑢 ∈ 𝐴 ⦋𝑢 / 𝑥⦌𝐵)
86	82, 85	xchnxbir 333	. . 3 ⊢ (¬ Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ≠ 𝑧 ∧ (𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) ≠ ∅))
87	61, 86	sylibr 234	. 2 ⊢ (¬ Disj 𝑦 ∈ ran 𝐹 𝑦 → ¬ Disj 𝑥 ∈ 𝐴 𝐵)
88	87	con4i 114	1 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑦 ∈ ran 𝐹 𝑦)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ∃wrex 3076  ⦋csb 3921   ∩ cin 3975  ∅c0 4352  Disj wdisj 5133   ↦ cmpt 5249  ran crn 5701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-disj 5134  df-br 5167  df-opab 5229  df-mpt 5250  df-cnv 5708  df-dm 5710  df-rn 5711

This theorem is referenced by:  meadjiun  46387