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Theorem el2mpocsbcl 7763
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 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))
Assertion
Ref Expression
el2mpocsbcl (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
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.
StepHypRef Expression
1 simpl 486 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → (𝑋𝐴𝑌𝐵))
2 el2mpocsbcl.o . . . . . . . . . . . . 13 𝑂 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))
3 nfcv 2955 . . . . . . . . . . . . . 14 𝑎(𝑠𝐶, 𝑡𝐷𝐸)
4 nfcv 2955 . . . . . . . . . . . . . 14 𝑏(𝑠𝐶, 𝑡𝐷𝐸)
5 nfcsb1v 3852 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐶
6 nfcsb1v 3852 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐷
7 nfcsb1v 3852 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐸
85, 6, 7nfmpo 7215 . . . . . . . . . . . . . 14 𝑥(𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸)
9 nfcv 2955 . . . . . . . . . . . . . . . 16 𝑦𝑎
10 nfcsb1v 3852 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐶
119, 10nfcsbw 3854 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐶
12 nfcsb1v 3852 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐷
139, 12nfcsbw 3854 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐷
14 nfcsb1v 3852 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐸
159, 14nfcsbw 3854 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐸
1611, 13, 15nfmpo 7215 . . . . . . . . . . . . . 14 𝑦(𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸)
17 csbeq1a 3842 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
18 csbeq1a 3842 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐶 = 𝑏 / 𝑦𝐶)
1918csbeq2dv 3835 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐶 = 𝑎 / 𝑥𝑏 / 𝑦𝐶)
2017, 19sylan9eq 2853 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐶 = 𝑎 / 𝑥𝑏 / 𝑦𝐶)
21 csbeq1a 3842 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐷 = 𝑎 / 𝑥𝐷)
22 csbeq1a 3842 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐷 = 𝑏 / 𝑦𝐷)
2322csbeq2dv 3835 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐷 = 𝑎 / 𝑥𝑏 / 𝑦𝐷)
2421, 23sylan9eq 2853 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐷 = 𝑎 / 𝑥𝑏 / 𝑦𝐷)
25 csbeq1a 3842 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐸 = 𝑎 / 𝑥𝐸)
26 csbeq1a 3842 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐸 = 𝑏 / 𝑦𝐸)
2726csbeq2dv 3835 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐸 = 𝑎 / 𝑥𝑏 / 𝑦𝐸)
2825, 27sylan9eq 2853 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐸 = 𝑎 / 𝑥𝑏 / 𝑦𝐸)
2920, 24, 28mpoeq123dv 7208 . . . . . . . . . . . . . 14 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑠𝐶, 𝑡𝐷𝐸) = (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸))
303, 4, 8, 16, 29cbvmpo 7227 . . . . . . . . . . . . 13 (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸)) = (𝑎𝐴, 𝑏𝐵 ↦ (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸))
312, 30eqtri 2821 . . . . . . . . . . . 12 𝑂 = (𝑎𝐴, 𝑏𝐵 ↦ (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸))
3231a1i 11 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑂 = (𝑎𝐴, 𝑏𝐵 ↦ (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸)))
33 csbeq1 3831 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑏 / 𝑦𝐶)
3433adantr 484 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑏 / 𝑦𝐶)
35 csbeq1 3831 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐶 = 𝑌 / 𝑦𝐶)
3635adantl 485 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐶 = 𝑌 / 𝑦𝐶)
3736csbeq2dv 3835 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑌 / 𝑦𝐶)
3834, 37eqtrd 2833 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑌 / 𝑦𝐶)
39 csbeq1 3831 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑏 / 𝑦𝐷)
4039adantr 484 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑏 / 𝑦𝐷)
41 csbeq1 3831 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐷 = 𝑌 / 𝑦𝐷)
4241adantl 485 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐷 = 𝑌 / 𝑦𝐷)
4342csbeq2dv 3835 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑌 / 𝑦𝐷)
4440, 43eqtrd 2833 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑌 / 𝑦𝐷)
45 csbeq1 3831 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑏 / 𝑦𝐸)
4645adantr 484 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑏 / 𝑦𝐸)
47 csbeq1 3831 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐸 = 𝑌 / 𝑦𝐸)
4847adantl 485 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐸 = 𝑌 / 𝑦𝐸)
4948csbeq2dv 3835 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑌 / 𝑦𝐸)
5046, 49eqtrd 2833 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑌 / 𝑦𝐸)
5138, 44, 50mpoeq123dv 7208 . . . . . . . . . . . 12 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸))
5251adantl 485 . . . . . . . . . . 11 (((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) ∧ (𝑎 = 𝑋𝑏 = 𝑌)) → (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸))
53 simpl 486 . . . . . . . . . . . 12 ((𝑋𝐴𝑌𝐵) → 𝑋𝐴)
5453adantl 485 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋𝐴)
55 simpr 488 . . . . . . . . . . . 12 ((𝑋𝐴𝑌𝐵) → 𝑌𝐵)
5655adantl 485 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑌𝐵)
57 simpl 486 . . . . . . . . . . . . . . . . . 18 ((𝐶𝑈𝐷𝑉) → 𝐶𝑈)
5857ralimi 3128 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑦𝐵 𝐶𝑈)
59 rspcsbela 4343 . . . . . . . . . . . . . . . . 17 ((𝑌𝐵 ∧ ∀𝑦𝐵 𝐶𝑈) → 𝑌 / 𝑦𝐶𝑈)
6055, 58, 59syl2an 598 . . . . . . . . . . . . . . . 16 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑦𝐵 (𝐶𝑈𝐷𝑉)) → 𝑌 / 𝑦𝐶𝑈)
6160ex 416 . . . . . . . . . . . . . . 15 ((𝑋𝐴𝑌𝐵) → (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → 𝑌 / 𝑦𝐶𝑈))
6261ralimdv 3145 . . . . . . . . . . . . . 14 ((𝑋𝐴𝑌𝐵) → (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈))
6362impcom 411 . . . . . . . . . . . . 13 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈)
64 rspcsbela 4343 . . . . . . . . . . . . 13 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈) → 𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈)
6554, 63, 64syl2anc 587 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈)
66 simpr 488 . . . . . . . . . . . . . . . . . 18 ((𝐶𝑈𝐷𝑉) → 𝐷𝑉)
6766ralimi 3128 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑦𝐵 𝐷𝑉)
68 rspcsbela 4343 . . . . . . . . . . . . . . . . 17 ((𝑌𝐵 ∧ ∀𝑦𝐵 𝐷𝑉) → 𝑌 / 𝑦𝐷𝑉)
6955, 67, 68syl2an 598 . . . . . . . . . . . . . . . 16 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑦𝐵 (𝐶𝑈𝐷𝑉)) → 𝑌 / 𝑦𝐷𝑉)
7069ex 416 . . . . . . . . . . . . . . 15 ((𝑋𝐴𝑌𝐵) → (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → 𝑌 / 𝑦𝐷𝑉))
7170ralimdv 3145 . . . . . . . . . . . . . 14 ((𝑋𝐴𝑌𝐵) → (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉))
7271impcom 411 . . . . . . . . . . . . 13 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉)
73 rspcsbela 4343 . . . . . . . . . . . . 13 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉) → 𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉)
7454, 72, 73syl2anc 587 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉)
75 mpoexga 7758 . . . . . . . . . . . 12 ((𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉) → (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) ∈ V)
7665, 74, 75syl2anc 587 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) ∈ V)
7732, 52, 54, 56, 76ovmpod 7281 . . . . . . . . . 10 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝑂𝑌) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸))
7877oveqd 7152 . . . . . . . . 9 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇))
7978eleq2d 2875 . . . . . . . 8 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) ↔ 𝑊 ∈ (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇)))
80 eqid 2798 . . . . . . . . 9 (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)
8180elmpocl 7367 . . . . . . . 8 (𝑊 ∈ (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))
8279, 81syl6bi 256 . . . . . . 7 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
8382impancom 455 . . . . . 6 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋𝐴𝑌𝐵) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
8483impcom 411 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))
851, 84jca 515 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
8685ex 416 . . 3 ((𝑋𝐴𝑌𝐵) → ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
872mpondm0 7366 . . . . . . 7 (¬ (𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = ∅)
8887oveqd 7152 . . . . . 6 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆𝑇))
8988eleq2d 2875 . . . . 5 (¬ (𝑋𝐴𝑌𝐵) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) ↔ 𝑊 ∈ (𝑆𝑇)))
90 noel 4247 . . . . . . 7 ¬ 𝑊 ∈ ∅
9190pm2.21i 119 . . . . . 6 (𝑊 ∈ ∅ → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
92 0ov 7172 . . . . . 6 (𝑆𝑇) = ∅
9391, 92eleq2s 2908 . . . . 5 (𝑊 ∈ (𝑆𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
9489, 93syl6bi 256 . . . 4 (¬ (𝑋𝐴𝑌𝐵) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
9594adantld 494 . . 3 (¬ (𝑋𝐴𝑌𝐵) → ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
9686, 95pm2.61i 185 . 2 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
9796ex 416 1 (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2111  wral 3106  Vcvv 3441  csb 3828  c0 4243  (class class class)co 7135  cmpo 7137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672
This theorem is referenced by:  el2mpocl  7764
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