Theorem el2mpocsbcl 8096

Description: If the operation value of the operation value of two nested maps-to notation is not empty, all involved arguments belong to the corresponding base classes of the maps-to notations. (Contributed by AV, 21-May-2021.)

Hypothesis

Ref	Expression
el2mpocsbcl.o	⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸))

Assertion

Ref	Expression
el2mpocsbcl	⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷))))

Distinct variable groups:   𝐴,𝑠,𝑡,𝑥,𝑦   𝐵,𝑠,𝑡,𝑥,𝑦   𝐶,𝑠,𝑡   𝐷,𝑠,𝑡   𝑥,𝑈,𝑦   𝑥,𝑉,𝑦   𝑋,𝑠,𝑡,𝑥,𝑦   𝑌,𝑠,𝑡,𝑥,𝑦

Allowed substitution hints:   𝐶(𝑥, 𝑦)   𝐷(𝑥, 𝑦)   𝑆(𝑥, 𝑦, 𝑡, 𝑠)   𝑇(𝑥, 𝑦, 𝑡, 𝑠)   𝑈(𝑡, 𝑠)   𝐸(𝑥, 𝑦, 𝑡, 𝑠)   𝑂(𝑥, 𝑦, 𝑡, 𝑠)   𝑉(𝑡, 𝑠)   𝑊(𝑥, 𝑦, 𝑡, 𝑠)

Proof of Theorem el2mpocsbcl

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 488	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵))
2		el2mpocsbcl.o	. . . . . . . . . . . . 13 ⊢ 𝑂 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸))
3		nfcv 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑎(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)
4		nfcv 2923	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏(𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)
5		nfcsb1v 3871	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶
6		nfcsb1v 3871	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷
7		nfcsb1v 3871	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸
8	5, 6, 7	nfmpo 7502	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝑠 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 ↦ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸)
9		nfcv 2923	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦𝑎
10		nfcsb1v 3871	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌𝐶
11	9, 10	nfcsbw 3873	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶
12		nfcsb1v 3871	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌𝐷
13	9, 12	nfcsbw 3873	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷
14		nfcsb1v 3871	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑦⦋𝑏 / 𝑦⦌𝐸
15	9, 14	nfcsbw 3873	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑦⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸
16	11, 13, 15	nfmpo 7502	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦(𝑠 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 ↦ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸)
17		csbeq1a 3861	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → 𝐶 = ⦋𝑎 / 𝑥⦌𝐶)
18		csbeq1a 3861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → 𝐶 = ⦋𝑏 / 𝑦⦌𝐶)
19	18	csbeq2dv 3854	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → ⦋𝑎 / 𝑥⦌𝐶 = ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶)
20	17, 19	sylan9eq 2816	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝐶 = ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶)
21		csbeq1a 3861	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → 𝐷 = ⦋𝑎 / 𝑥⦌𝐷)
22		csbeq1a 3861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → 𝐷 = ⦋𝑏 / 𝑦⦌𝐷)
23	22	csbeq2dv 3854	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → ⦋𝑎 / 𝑥⦌𝐷 = ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷)
24	21, 23	sylan9eq 2816	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝐷 = ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷)
25		csbeq1a 3861	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑎 → 𝐸 = ⦋𝑎 / 𝑥⦌𝐸)
26		csbeq1a 3861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑏 → 𝐸 = ⦋𝑏 / 𝑦⦌𝐸)
27	26	csbeq2dv 3854	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑏 → ⦋𝑎 / 𝑥⦌𝐸 = ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸)
28	25, 27	sylan9eq 2816	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → 𝐸 = ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸)
29	20, 24, 28	mpoeq123dv 7495	. . . . . . . . . . . . . 14 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑏) → (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸) = (𝑠 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 ↦ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸))
30	3, 4, 8, 16, 29	cbvmpo 7514	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ (𝑠 ∈ 𝐶, 𝑡 ∈ 𝐷 ↦ 𝐸)) = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ (𝑠 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 ↦ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸))
31	2, 30	eqtri 2784	. . . . . . . . . . . 12 ⊢ 𝑂 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ (𝑠 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 ↦ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸))
32	31	a1i 11	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝑂 = (𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵 ↦ (𝑠 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 ↦ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸)))
33		csbeq1 3850	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑋 → ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶 = ⦋𝑋 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶)
34	33	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶 = ⦋𝑋 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶)
35		csbeq1 3850	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑌 → ⦋𝑏 / 𝑦⦌𝐶 = ⦋𝑌 / 𝑦⦌𝐶)
36	35	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑏 / 𝑦⦌𝐶 = ⦋𝑌 / 𝑦⦌𝐶)
37	36	csbeq2dv 3854	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑋 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶 = ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶)
38	34, 37	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶 = ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶)
39		csbeq1 3850	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑋 → ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 = ⦋𝑋 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷)
40	39	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 = ⦋𝑋 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷)
41		csbeq1 3850	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑌 → ⦋𝑏 / 𝑦⦌𝐷 = ⦋𝑌 / 𝑦⦌𝐷)
42	41	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑏 / 𝑦⦌𝐷 = ⦋𝑌 / 𝑦⦌𝐷)
43	42	csbeq2dv 3854	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑋 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 = ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)
44	40, 43	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 = ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)
45		csbeq1 3850	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑋 → ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸 = ⦋𝑋 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸)
46	45	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸 = ⦋𝑋 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸)
47		csbeq1 3850	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = 𝑌 → ⦋𝑏 / 𝑦⦌𝐸 = ⦋𝑌 / 𝑦⦌𝐸)
48	47	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑏 / 𝑦⦌𝐸 = ⦋𝑌 / 𝑦⦌𝐸)
49	48	csbeq2dv 3854	. . . . . . . . . . . . . 14 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑋 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸 = ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸)
50	46, 49	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸 = ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸)
51	38, 44, 50	mpoeq123dv 7495	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝑋 ∧ 𝑏 = 𝑌) → (𝑠 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 ↦ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸) = (𝑠 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↦ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸))
52	51	adantl 487	. . . . . . . . . . 11 ⊢ (((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑎 = 𝑋 ∧ 𝑏 = 𝑌)) → (𝑠 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐷 ↦ ⦋𝑎 / 𝑥⦌⦋𝑏 / 𝑦⦌𝐸) = (𝑠 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↦ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸))
53		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑋 ∈ 𝐴)
54	53	adantl 487	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐴)
55		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ 𝐵)
56	55	adantl 487	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
57		simpl 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → 𝐶 ∈ 𝑈)
58	57	ralimi 3100	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑈)
59		rspcsbela 4396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 𝐶 ∈ 𝑈) → ⦋𝑌 / 𝑦⦌𝐶 ∈ 𝑈)
60	55, 58, 59	syl2an 608	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉)) → ⦋𝑌 / 𝑦⦌𝐶 ∈ 𝑈)
61	60	ex 418	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → ⦋𝑌 / 𝑦⦌𝐶 ∈ 𝑈))
62	61	ralimdv 3177	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → ∀𝑥 ∈ 𝐴 ⦋𝑌 / 𝑦⦌𝐶 ∈ 𝑈))
63	62	impcom 413	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋𝑌 / 𝑦⦌𝐶 ∈ 𝑈)
64		rspcsbela 4396	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋𝑌 / 𝑦⦌𝐶 ∈ 𝑈) → ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∈ 𝑈)
65	54, 63, 64	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∈ 𝑈)
66		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → 𝐷 ∈ 𝑉)
67	66	ralimi 3100	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → ∀𝑦 ∈ 𝐵 𝐷 ∈ 𝑉)
68		rspcsbela 4396	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑌 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 𝐷 ∈ 𝑉) → ⦋𝑌 / 𝑦⦌𝐷 ∈ 𝑉)
69	55, 67, 68	syl2an 608	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉)) → ⦋𝑌 / 𝑦⦌𝐷 ∈ 𝑉)
70	69	ex 418	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → ⦋𝑌 / 𝑦⦌𝐷 ∈ 𝑉))
71	70	ralimdv 3177	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → ∀𝑥 ∈ 𝐴 ⦋𝑌 / 𝑦⦌𝐷 ∈ 𝑉))
72	71	impcom 413	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 ⦋𝑌 / 𝑦⦌𝐷 ∈ 𝑉)
73		rspcsbela 4396	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ⦋𝑌 / 𝑦⦌𝐷 ∈ 𝑉) → ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ∈ 𝑉)
74	54, 72, 73	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ∈ 𝑉)
75		mpoexga 8090	. . . . . . . . . . . 12 ⊢ ((⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∈ 𝑈 ∧ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ∈ 𝑉) → (𝑠 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↦ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸) ∈ V)
76	65, 74, 75	syl2anc 596	. . . . . . . . . . 11 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (𝑠 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↦ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸) ∈ V)
77	32, 52, 54, 56, 76	ovmpod 7572	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝑂𝑌) = (𝑠 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↦ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸))
78	77	oveqd 7437	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↦ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸)𝑇))
79	78	eleq2d 2847	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) ↔ 𝑊 ∈ (𝑆(𝑠 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↦ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸)𝑇)))
80		eqid 2761	. . . . . . . . 9 ⊢ (𝑠 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↦ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸) = (𝑠 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↦ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸)
81	80	elmpocl 7662	. . . . . . . 8 ⊢ (𝑊 ∈ (𝑆(𝑠 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶, 𝑡 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷 ↦ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐸)𝑇) → (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷))
82	79, 81	biimtrdi 256	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵)) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)))
83	82	impancom 457	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)))
84	83	impcom 413	. . . . 5 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷))
85	1, 84	jca 521	. . . 4 ⊢ (((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)))
86	85	ex 418	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷))))
87	2	mpondm0 7661	. . . . . . 7 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑋𝑂𝑌) = ∅)
88	87	oveqd 7437	. . . . . 6 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆∅𝑇))
89	88	eleq2d 2847	. . . . 5 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) ↔ 𝑊 ∈ (𝑆∅𝑇)))
90		noel 4284	. . . . . . 7 ⊢ ¬ 𝑊 ∈ ∅
91	90	pm2.21i 120	. . . . . 6 ⊢ (𝑊 ∈ ∅ → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)))
92		0ov 7457	. . . . . 6 ⊢ (𝑆∅𝑇) = ∅
93	91, 92	eleq2s 2879	. . . . 5 ⊢ (𝑊 ∈ (𝑆∅𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)))
94	89, 93	biimtrdi 256	. . . 4 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷))))
95	94	adantld 496	. . 3 ⊢ (¬ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷))))
96	86, 95	pm2.61i 184	. 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷)))
97	96	ex 418	1 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝐶 ∈ 𝑈 ∧ 𝐷 ∈ 𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) ∧ (𝑆 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐶 ∧ 𝑇 ∈ ⦋𝑋 / 𝑥⦌⦋𝑌 / 𝑦⦌𝐷))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  Vcvv 3451  ⦋csb 3847  ∅c0 4279  (class class class)co 7420   ∈ cmpo 7422

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002

This theorem is used by:  el2mpocl  8097