Theorem xpassen 6832

Description: Associative law for equinumerosity of Cartesian product. Proposition 4.22(e) of [Mendelson] p. 254. (Contributed by NM, 22-Jan-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)

Hypotheses

Ref	Expression
xpassen.1	⊢ 𝐴 ∈ V
xpassen.2	⊢ 𝐵 ∈ V
xpassen.3	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
xpassen	⊢ ((𝐴 × 𝐵) × 𝐶) ≈ (𝐴 × (𝐵 × 𝐶))

Proof of Theorem xpassen

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

Step	Hyp	Ref	Expression
1		xpassen.1	. . . 4 ⊢ 𝐴 ∈ V
2		xpassen.2	. . . 4 ⊢ 𝐵 ∈ V
3	1, 2	xpex 4743	. . 3 ⊢ (𝐴 × 𝐵) ∈ V
4		xpassen.3	. . 3 ⊢ 𝐶 ∈ V
5	3, 4	xpex 4743	. 2 ⊢ ((𝐴 × 𝐵) × 𝐶) ∈ V
6	2, 4	xpex 4743	. . 3 ⊢ (𝐵 × 𝐶) ∈ V
7	1, 6	xpex 4743	. 2 ⊢ (𝐴 × (𝐵 × 𝐶)) ∈ V
8		vex 2742	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
9	8	snex 4187	. . . . . . . . 9 ⊢ {𝑥} ∈ V
10	9	dmex 4895	. . . . . . . 8 ⊢ dom {𝑥} ∈ V
11	10	uniex 4439	. . . . . . 7 ⊢ ∪ dom {𝑥} ∈ V
12	11	snex 4187	. . . . . 6 ⊢ {∪ dom {𝑥}} ∈ V
13	12	dmex 4895	. . . . 5 ⊢ dom {∪ dom {𝑥}} ∈ V
14	13	uniex 4439	. . . 4 ⊢ ∪ dom {∪ dom {𝑥}} ∈ V
15	12	rnex 4896	. . . . . 6 ⊢ ran {∪ dom {𝑥}} ∈ V
16	15	uniex 4439	. . . . 5 ⊢ ∪ ran {∪ dom {𝑥}} ∈ V
17	9	rnex 4896	. . . . . 6 ⊢ ran {𝑥} ∈ V
18	17	uniex 4439	. . . . 5 ⊢ ∪ ran {𝑥} ∈ V
19	16, 18	opex 4231	. . . 4 ⊢ ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩ ∈ V
20	14, 19	opex 4231	. . 3 ⊢ ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩ ∈ V
21	20	a1i 9	. 2 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) → ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩ ∈ V)
22		vex 2742	. . . . . . . 8 ⊢ 𝑦 ∈ V
23	22	snex 4187	. . . . . . 7 ⊢ {𝑦} ∈ V
24	23	dmex 4895	. . . . . 6 ⊢ dom {𝑦} ∈ V
25	24	uniex 4439	. . . . 5 ⊢ ∪ dom {𝑦} ∈ V
26	23	rnex 4896	. . . . . . . . 9 ⊢ ran {𝑦} ∈ V
27	26	uniex 4439	. . . . . . . 8 ⊢ ∪ ran {𝑦} ∈ V
28	27	snex 4187	. . . . . . 7 ⊢ {∪ ran {𝑦}} ∈ V
29	28	dmex 4895	. . . . . 6 ⊢ dom {∪ ran {𝑦}} ∈ V
30	29	uniex 4439	. . . . 5 ⊢ ∪ dom {∪ ran {𝑦}} ∈ V
31	25, 30	opex 4231	. . . 4 ⊢ ⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩ ∈ V
32	28	rnex 4896	. . . . 5 ⊢ ran {∪ ran {𝑦}} ∈ V
33	32	uniex 4439	. . . 4 ⊢ ∪ ran {∪ ran {𝑦}} ∈ V
34	31, 33	opex 4231	. . 3 ⊢ ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩ ∈ V
35	34	a1i 9	. 2 ⊢ (𝑦 ∈ (𝐴 × (𝐵 × 𝐶)) → ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩ ∈ V)
36		sneq 3605	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → {𝑥} = {⟨⟨𝑧, 𝑤⟩, 𝑣⟩})
37	36	dmeqd 4831	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → dom {𝑥} = dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩})
38	37	unieqd 3822	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → ∪ dom {𝑥} = ∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩})
39	38	sneqd 3607	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → {∪ dom {𝑥}} = {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}})
40	39	dmeqd 4831	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → dom {∪ dom {𝑥}} = dom {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}})
41	40	unieqd 3822	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → ∪ dom {∪ dom {𝑥}} = ∪ dom {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}})
42		vex 2742	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑧 ∈ V
43		vex 2742	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
44	42, 43	opex 4231	. . . . . . . . . . . . . . . . 17 ⊢ ⟨𝑧, 𝑤⟩ ∈ V
45		vex 2742	. . . . . . . . . . . . . . . . 17 ⊢ 𝑣 ∈ V
46	44, 45	op1sta 5112	. . . . . . . . . . . . . . . 16 ⊢ ∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩} = ⟨𝑧, 𝑤⟩
47	46	sneqi 3606	. . . . . . . . . . . . . . 15 ⊢ {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}} = {⟨𝑧, 𝑤⟩}
48	47	dmeqi 4830	. . . . . . . . . . . . . 14 ⊢ dom {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}} = dom {⟨𝑧, 𝑤⟩}
49	48	unieqi 3821	. . . . . . . . . . . . 13 ⊢ ∪ dom {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}} = ∪ dom {⟨𝑧, 𝑤⟩}
50	42, 43	op1sta 5112	. . . . . . . . . . . . 13 ⊢ ∪ dom {⟨𝑧, 𝑤⟩} = 𝑧
51	49, 50	eqtri 2198	. . . . . . . . . . . 12 ⊢ ∪ dom {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}} = 𝑧
52	41, 51	eqtr2di 2227	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → 𝑧 = ∪ dom {∪ dom {𝑥}})
53	39	rneqd 4858	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → ran {∪ dom {𝑥}} = ran {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}})
54	53	unieqd 3822	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → ∪ ran {∪ dom {𝑥}} = ∪ ran {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}})
55	47	rneqi 4857	. . . . . . . . . . . . . . 15 ⊢ ran {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}} = ran {⟨𝑧, 𝑤⟩}
56	55	unieqi 3821	. . . . . . . . . . . . . 14 ⊢ ∪ ran {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}} = ∪ ran {⟨𝑧, 𝑤⟩}
57	42, 43	op2nda 5115	. . . . . . . . . . . . . 14 ⊢ ∪ ran {⟨𝑧, 𝑤⟩} = 𝑤
58	56, 57	eqtri 2198	. . . . . . . . . . . . 13 ⊢ ∪ ran {∪ dom {⟨⟨𝑧, 𝑤⟩, 𝑣⟩}} = 𝑤
59	54, 58	eqtr2di 2227	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → 𝑤 = ∪ ran {∪ dom {𝑥}})
60	36	rneqd 4858	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → ran {𝑥} = ran {⟨⟨𝑧, 𝑤⟩, 𝑣⟩})
61	60	unieqd 3822	. . . . . . . . . . . . 13 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → ∪ ran {𝑥} = ∪ ran {⟨⟨𝑧, 𝑤⟩, 𝑣⟩})
62	44, 45	op2nda 5115	. . . . . . . . . . . . 13 ⊢ ∪ ran {⟨⟨𝑧, 𝑤⟩, 𝑣⟩} = 𝑣
63	61, 62	eqtr2di 2227	. . . . . . . . . . . 12 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → 𝑣 = ∪ ran {𝑥})
64	59, 63	opeq12d 3788	. . . . . . . . . . 11 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → ⟨𝑤, 𝑣⟩ = ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩)
65	52, 64	opeq12d 3788	. . . . . . . . . 10 ⊢ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ → ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩)
66		sneq 3605	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → {𝑦} = {⟨𝑧, ⟨𝑤, 𝑣⟩⟩})
67	66	dmeqd 4831	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → dom {𝑦} = dom {⟨𝑧, ⟨𝑤, 𝑣⟩⟩})
68	67	unieqd 3822	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → ∪ dom {𝑦} = ∪ dom {⟨𝑧, ⟨𝑤, 𝑣⟩⟩})
69	43, 45	opex 4231	. . . . . . . . . . . . . 14 ⊢ ⟨𝑤, 𝑣⟩ ∈ V
70	42, 69	op1sta 5112	. . . . . . . . . . . . 13 ⊢ ∪ dom {⟨𝑧, ⟨𝑤, 𝑣⟩⟩} = 𝑧
71	68, 70	eqtr2di 2227	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → 𝑧 = ∪ dom {𝑦})
72	66	rneqd 4858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → ran {𝑦} = ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩})
73	72	unieqd 3822	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → ∪ ran {𝑦} = ∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩})
74	73	sneqd 3607	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → {∪ ran {𝑦}} = {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}})
75	74	dmeqd 4831	. . . . . . . . . . . . . 14 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → dom {∪ ran {𝑦}} = dom {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}})
76	75	unieqd 3822	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → ∪ dom {∪ ran {𝑦}} = ∪ dom {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}})
77	42, 69	op2nda 5115	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩} = ⟨𝑤, 𝑣⟩
78	77	sneqi 3606	. . . . . . . . . . . . . . . 16 ⊢ {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}} = {⟨𝑤, 𝑣⟩}
79	78	dmeqi 4830	. . . . . . . . . . . . . . 15 ⊢ dom {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}} = dom {⟨𝑤, 𝑣⟩}
80	79	unieqi 3821	. . . . . . . . . . . . . 14 ⊢ ∪ dom {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}} = ∪ dom {⟨𝑤, 𝑣⟩}
81	43, 45	op1sta 5112	. . . . . . . . . . . . . 14 ⊢ ∪ dom {⟨𝑤, 𝑣⟩} = 𝑤
82	80, 81	eqtri 2198	. . . . . . . . . . . . 13 ⊢ ∪ dom {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}} = 𝑤
83	76, 82	eqtr2di 2227	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → 𝑤 = ∪ dom {∪ ran {𝑦}})
84	71, 83	opeq12d 3788	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → ⟨𝑧, 𝑤⟩ = ⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩)
85	74	rneqd 4858	. . . . . . . . . . . . 13 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → ran {∪ ran {𝑦}} = ran {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}})
86	85	unieqd 3822	. . . . . . . . . . . 12 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → ∪ ran {∪ ran {𝑦}} = ∪ ran {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}})
87	78	rneqi 4857	. . . . . . . . . . . . . 14 ⊢ ran {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}} = ran {⟨𝑤, 𝑣⟩}
88	87	unieqi 3821	. . . . . . . . . . . . 13 ⊢ ∪ ran {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}} = ∪ ran {⟨𝑤, 𝑣⟩}
89	43, 45	op2nda 5115	. . . . . . . . . . . . 13 ⊢ ∪ ran {⟨𝑤, 𝑣⟩} = 𝑣
90	88, 89	eqtri 2198	. . . . . . . . . . . 12 ⊢ ∪ ran {∪ ran {⟨𝑧, ⟨𝑤, 𝑣⟩⟩}} = 𝑣
91	86, 90	eqtr2di 2227	. . . . . . . . . . 11 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → 𝑣 = ∪ ran {∪ ran {𝑦}})
92	84, 91	opeq12d 3788	. . . . . . . . . 10 ⊢ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ → ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩)
93	65, 92	eq2tri 2237	. . . . . . . . 9 ⊢ ((𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩))
94		anass 401	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶) ↔ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))
95	93, 94	anbi12i 460	. . . . . . . 8 ⊢ (((𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ↔ ((𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩) ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
96		an32 562	. . . . . . . 8 ⊢ (((𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ ((𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)))
97		an32 562	. . . . . . . 8 ⊢ (((𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩) ↔ ((𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩) ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
98	95, 96, 97	3bitr4i 212	. . . . . . 7 ⊢ (((𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ ((𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩))
99	98	exbii 1605	. . . . . 6 ⊢ (∃𝑣((𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ ∃𝑣((𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩))
100		19.41v 1902	. . . . . 6 ⊢ (∃𝑣((𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ (∃𝑣(𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩))
101		19.41v 1902	. . . . . 6 ⊢ (∃𝑣((𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩) ↔ (∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩))
102	99, 100, 101	3bitr3i 210	. . . . 5 ⊢ ((∃𝑣(𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ (∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩))
103	102	2exbii 1606	. . . 4 ⊢ (∃𝑧∃𝑤(∃𝑣(𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ ∃𝑧∃𝑤(∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩))
104		19.41vv 1903	. . . 4 ⊢ (∃𝑧∃𝑤(∃𝑣(𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ (∃𝑧∃𝑤∃𝑣(𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩))
105		19.41vv 1903	. . . 4 ⊢ (∃𝑧∃𝑤(∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩) ↔ (∃𝑧∃𝑤∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩))
106	103, 104, 105	3bitr3i 210	. . 3 ⊢ ((∃𝑧∃𝑤∃𝑣(𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ (∃𝑧∃𝑤∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩))
107		elxp 4645	. . . . 5 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑢∃𝑣(𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑣 ∈ 𝐶)))
108		excom 1664	. . . . 5 ⊢ (∃𝑢∃𝑣(𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑣 ∈ 𝐶)) ↔ ∃𝑣∃𝑢(𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑣 ∈ 𝐶)))
109		elxp 4645	. . . . . . . . 9 ⊢ (𝑢 ∈ (𝐴 × 𝐵) ↔ ∃𝑧∃𝑤(𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)))
110	109	anbi1i 458	. . . . . . . 8 ⊢ ((𝑢 ∈ (𝐴 × 𝐵) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)) ↔ (∃𝑧∃𝑤(𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)))
111		an12 561	. . . . . . . 8 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑣 ∈ 𝐶)) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)))
112		19.41vv 1903	. . . . . . . 8 ⊢ (∃𝑧∃𝑤((𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)) ↔ (∃𝑧∃𝑤(𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)))
113	110, 111, 112	3bitr4i 212	. . . . . . 7 ⊢ ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑣 ∈ 𝐶)) ↔ ∃𝑧∃𝑤((𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)))
114	113	2exbii 1606	. . . . . 6 ⊢ (∃𝑣∃𝑢(𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑣 ∈ 𝐶)) ↔ ∃𝑣∃𝑢∃𝑧∃𝑤((𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)))
115		exrot4 1691	. . . . . 6 ⊢ (∃𝑣∃𝑢∃𝑧∃𝑤((𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)))
116		anass 401	. . . . . . . . 9 ⊢ (((𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)) ↔ (𝑢 = ⟨𝑧, 𝑤⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶))))
117	116	exbii 1605	. . . . . . . 8 ⊢ (∃𝑢((𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)) ↔ ∃𝑢(𝑢 = ⟨𝑧, 𝑤⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶))))
118		opeq1 3780	. . . . . . . . . . . 12 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ⟨𝑢, 𝑣⟩ = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩)
119	118	eqeq2d 2189	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → (𝑥 = ⟨𝑢, 𝑣⟩ ↔ 𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩))
120	119	anbi1d 465	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → ((𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶) ↔ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)))
121	120	anbi2d 464	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑧, 𝑤⟩ → (((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ 𝑣 ∈ 𝐶))))
122	44, 121	ceqsexv 2778	. . . . . . . 8 ⊢ (∃𝑢(𝑢 = ⟨𝑧, 𝑤⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶))) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)))
123		an12 561	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)) ↔ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)))
124	117, 122, 123	3bitri 206	. . . . . . 7 ⊢ (∃𝑢((𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)) ↔ (𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)))
125	124	3exbii 1607	. . . . . 6 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢((𝑢 = ⟨𝑧, 𝑤⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) ∧ (𝑥 = ⟨𝑢, 𝑣⟩ ∧ 𝑣 ∈ 𝐶)) ↔ ∃𝑧∃𝑤∃𝑣(𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)))
126	114, 115, 125	3bitri 206	. . . . 5 ⊢ (∃𝑣∃𝑢(𝑥 = ⟨𝑢, 𝑣⟩ ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑣 ∈ 𝐶)) ↔ ∃𝑧∃𝑤∃𝑣(𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)))
127	107, 108, 126	3bitri 206	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ↔ ∃𝑧∃𝑤∃𝑣(𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)))
128	127	anbi1i 458	. . 3 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ (∃𝑧∃𝑤∃𝑣(𝑥 = ⟨⟨𝑧, 𝑤⟩, 𝑣⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ∧ 𝑣 ∈ 𝐶)) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩))
129		elxp 4645	. . . . 5 ⊢ (𝑦 ∈ (𝐴 × (𝐵 × 𝐶)) ↔ ∃𝑧∃𝑢(𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑢 ∈ (𝐵 × 𝐶))))
130		elxp 4645	. . . . . . . . . 10 ⊢ (𝑢 ∈ (𝐵 × 𝐶) ↔ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))
131	130	anbi2i 457	. . . . . . . . 9 ⊢ (((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ 𝑢 ∈ (𝐵 × 𝐶)) ↔ ((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
132		anass 401	. . . . . . . . 9 ⊢ (((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ 𝑢 ∈ (𝐵 × 𝐶)) ↔ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑢 ∈ (𝐵 × 𝐶))))
133		19.42vv 1911	. . . . . . . . . 10 ⊢ (∃𝑤∃𝑣((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ↔ ((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
134		an12 561	. . . . . . . . . . . 12 ⊢ (((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ↔ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ ((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
135		anass 401	. . . . . . . . . . . . 13 ⊢ (((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)) ↔ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
136	135	anbi2i 457	. . . . . . . . . . . 12 ⊢ ((𝑢 = ⟨𝑤, 𝑣⟩ ∧ ((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ↔ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))))
137	134, 136	bitri 184	. . . . . . . . . . 11 ⊢ (((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ↔ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))))
138	137	2exbii 1606	. . . . . . . . . 10 ⊢ (∃𝑤∃𝑣((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ (𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ↔ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))))
139	133, 138	bitr3i 186	. . . . . . . . 9 ⊢ (((𝑦 = ⟨𝑧, 𝑢⟩ ∧ 𝑧 ∈ 𝐴) ∧ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ↔ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))))
140	131, 132, 139	3bitr3i 210	. . . . . . . 8 ⊢ ((𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑢 ∈ (𝐵 × 𝐶))) ↔ ∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))))
141	140	exbii 1605	. . . . . . 7 ⊢ (∃𝑢(𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑢 ∈ (𝐵 × 𝐶))) ↔ ∃𝑢∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))))
142		exrot3 1690	. . . . . . 7 ⊢ (∃𝑢∃𝑤∃𝑣(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))) ↔ ∃𝑤∃𝑣∃𝑢(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))))
143		opeq2 3781	. . . . . . . . . . 11 ⊢ (𝑢 = ⟨𝑤, 𝑣⟩ → ⟨𝑧, 𝑢⟩ = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩)
144	143	eqeq2d 2189	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑤, 𝑣⟩ → (𝑦 = ⟨𝑧, 𝑢⟩ ↔ 𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩))
145	144	anbi1d 465	. . . . . . . . 9 ⊢ (𝑢 = ⟨𝑤, 𝑣⟩ → ((𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ↔ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))))
146	69, 145	ceqsexv 2778	. . . . . . . 8 ⊢ (∃𝑢(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))) ↔ (𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
147	146	2exbii 1606	. . . . . . 7 ⊢ (∃𝑤∃𝑣∃𝑢(𝑢 = ⟨𝑤, 𝑣⟩ ∧ (𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶)))) ↔ ∃𝑤∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
148	141, 142, 147	3bitri 206	. . . . . 6 ⊢ (∃𝑢(𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑢 ∈ (𝐵 × 𝐶))) ↔ ∃𝑤∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
149	148	exbii 1605	. . . . 5 ⊢ (∃𝑧∃𝑢(𝑦 = ⟨𝑧, 𝑢⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑢 ∈ (𝐵 × 𝐶))) ↔ ∃𝑧∃𝑤∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
150	129, 149	bitri 184	. . . 4 ⊢ (𝑦 ∈ (𝐴 × (𝐵 × 𝐶)) ↔ ∃𝑧∃𝑤∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))))
151	150	anbi1i 458	. . 3 ⊢ ((𝑦 ∈ (𝐴 × (𝐵 × 𝐶)) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩) ↔ (∃𝑧∃𝑤∃𝑣(𝑦 = ⟨𝑧, ⟨𝑤, 𝑣⟩⟩ ∧ (𝑧 ∈ 𝐴 ∧ (𝑤 ∈ 𝐵 ∧ 𝑣 ∈ 𝐶))) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩))
152	106, 128, 151	3bitr4i 212	. 2 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) × 𝐶) ∧ 𝑦 = ⟨∪ dom {∪ dom {𝑥}}, ⟨∪ ran {∪ dom {𝑥}}, ∪ ran {𝑥}⟩⟩) ↔ (𝑦 ∈ (𝐴 × (𝐵 × 𝐶)) ∧ 𝑥 = ⟨⟨∪ dom {𝑦}, ∪ dom {∪ ran {𝑦}}⟩, ∪ ran {∪ ran {𝑦}}⟩))
153	5, 7, 21, 35, 152	en2i 6772	1 ⊢ ((𝐴 × 𝐵) × 𝐶) ≈ (𝐴 × (𝐵 × 𝐶))

Syntax hints:   ∧ wa 104   = wceq 1353  ∃wex 1492   ∈ wcel 2148  Vcvv 2739  {csn 3594  ⟨cop 3597  ∪ cuni 3811   class class class wbr 4005   × cxp 4626  dom cdm 4628  ran crn 4629   ≈ cen 6740

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-en 6743

This theorem is referenced by: (None)