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Theorem el2mpt2csbcl 7137
 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
el2mpt2csbcl.o 𝑂 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))
Assertion
Ref Expression
el2mpt2csbcl (∀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 el2mpt2csbcl
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 472 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → (𝑋𝐴𝑌𝐵))
2 el2mpt2csbcl.o . . . . . . . . . . . . 13 𝑂 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))
3 nfcv 2751 . . . . . . . . . . . . . 14 𝑎(𝑠𝐶, 𝑡𝐷𝐸)
4 nfcv 2751 . . . . . . . . . . . . . 14 𝑏(𝑠𝐶, 𝑡𝐷𝐸)
5 nfcsb1v 3515 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐶
6 nfcsb1v 3515 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐷
7 nfcsb1v 3515 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐸
85, 6, 7nfmpt2 6622 . . . . . . . . . . . . . 14 𝑥(𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸)
9 nfcv 2751 . . . . . . . . . . . . . . . 16 𝑦𝑎
10 nfcsb1v 3515 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐶
119, 10nfcsb 3517 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐶
12 nfcsb1v 3515 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐷
139, 12nfcsb 3517 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐷
14 nfcsb1v 3515 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐸
159, 14nfcsb 3517 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐸
1611, 13, 15nfmpt2 6622 . . . . . . . . . . . . . 14 𝑦(𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸)
17 csbeq1a 3508 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
18 csbeq1a 3508 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐶 = 𝑏 / 𝑦𝐶)
1918csbeq2dv 3944 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐶 = 𝑎 / 𝑥𝑏 / 𝑦𝐶)
2017, 19sylan9eq 2664 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐶 = 𝑎 / 𝑥𝑏 / 𝑦𝐶)
21 csbeq1a 3508 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐷 = 𝑎 / 𝑥𝐷)
22 csbeq1a 3508 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐷 = 𝑏 / 𝑦𝐷)
2322csbeq2dv 3944 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐷 = 𝑎 / 𝑥𝑏 / 𝑦𝐷)
2421, 23sylan9eq 2664 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐷 = 𝑎 / 𝑥𝑏 / 𝑦𝐷)
25 csbeq1a 3508 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐸 = 𝑎 / 𝑥𝐸)
26 csbeq1a 3508 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐸 = 𝑏 / 𝑦𝐸)
2726csbeq2dv 3944 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐸 = 𝑎 / 𝑥𝑏 / 𝑦𝐸)
2825, 27sylan9eq 2664 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐸 = 𝑎 / 𝑥𝑏 / 𝑦𝐸)
2920, 24, 28mpt2eq123dv 6615 . . . . . . . . . . . . . 14 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑠𝐶, 𝑡𝐷𝐸) = (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸))
303, 4, 8, 16, 29cbvmpt2 6632 . . . . . . . . . . . . 13 (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸)) = (𝑎𝐴, 𝑏𝐵 ↦ (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸))
312, 30eqtri 2632 . . . . . . . . . . . 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 3502 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑏 / 𝑦𝐶)
3433adantr 480 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑏 / 𝑦𝐶)
35 csbeq1 3502 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐶 = 𝑌 / 𝑦𝐶)
3635adantl 481 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐶 = 𝑌 / 𝑦𝐶)
3736csbeq2dv 3944 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑌 / 𝑦𝐶)
3834, 37eqtrd 2644 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑌 / 𝑦𝐶)
39 csbeq1 3502 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑏 / 𝑦𝐷)
4039adantr 480 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑏 / 𝑦𝐷)
41 csbeq1 3502 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐷 = 𝑌 / 𝑦𝐷)
4241adantl 481 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐷 = 𝑌 / 𝑦𝐷)
4342csbeq2dv 3944 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑌 / 𝑦𝐷)
4440, 43eqtrd 2644 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑌 / 𝑦𝐷)
45 csbeq1 3502 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑏 / 𝑦𝐸)
4645adantr 480 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑏 / 𝑦𝐸)
47 csbeq1 3502 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐸 = 𝑌 / 𝑦𝐸)
4847adantl 481 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐸 = 𝑌 / 𝑦𝐸)
4948csbeq2dv 3944 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑌 / 𝑦𝐸)
5046, 49eqtrd 2644 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑌 / 𝑦𝐸)
5138, 44, 50mpt2eq123dv 6615 . . . . . . . . . . . 12 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸))
5251adantl 481 . . . . . . . . . . 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 472 . . . . . . . . . . . 12 ((𝑋𝐴𝑌𝐵) → 𝑋𝐴)
5453adantl 481 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋𝐴)
55 simpr 476 . . . . . . . . . . . 12 ((𝑋𝐴𝑌𝐵) → 𝑌𝐵)
5655adantl 481 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑌𝐵)
57 simpl 472 . . . . . . . . . . . . . . . . . 18 ((𝐶𝑈𝐷𝑉) → 𝐶𝑈)
5857ralimi 2936 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑦𝐵 𝐶𝑈)
59 rspcsbela 3958 . . . . . . . . . . . . . . . . 17 ((𝑌𝐵 ∧ ∀𝑦𝐵 𝐶𝑈) → 𝑌 / 𝑦𝐶𝑈)
6055, 58, 59syl2an 493 . . . . . . . . . . . . . . . 16 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑦𝐵 (𝐶𝑈𝐷𝑉)) → 𝑌 / 𝑦𝐶𝑈)
6160ex 449 . . . . . . . . . . . . . . 15 ((𝑋𝐴𝑌𝐵) → (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → 𝑌 / 𝑦𝐶𝑈))
6261ralimdv 2946 . . . . . . . . . . . . . 14 ((𝑋𝐴𝑌𝐵) → (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈))
6362impcom 445 . . . . . . . . . . . . 13 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈)
64 rspcsbela 3958 . . . . . . . . . . . . 13 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈) → 𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈)
6554, 63, 64syl2anc 691 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈)
66 simpr 476 . . . . . . . . . . . . . . . . . 18 ((𝐶𝑈𝐷𝑉) → 𝐷𝑉)
6766ralimi 2936 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑦𝐵 𝐷𝑉)
68 rspcsbela 3958 . . . . . . . . . . . . . . . . 17 ((𝑌𝐵 ∧ ∀𝑦𝐵 𝐷𝑉) → 𝑌 / 𝑦𝐷𝑉)
6955, 67, 68syl2an 493 . . . . . . . . . . . . . . . 16 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑦𝐵 (𝐶𝑈𝐷𝑉)) → 𝑌 / 𝑦𝐷𝑉)
7069ex 449 . . . . . . . . . . . . . . 15 ((𝑋𝐴𝑌𝐵) → (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → 𝑌 / 𝑦𝐷𝑉))
7170ralimdv 2946 . . . . . . . . . . . . . 14 ((𝑋𝐴𝑌𝐵) → (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉))
7271impcom 445 . . . . . . . . . . . . 13 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉)
73 rspcsbela 3958 . . . . . . . . . . . . 13 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉) → 𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉)
7454, 72, 73syl2anc 691 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉)
75 mpt2exga 7135 . . . . . . . . . . . 12 ((𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉) → (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) ∈ V)
7665, 74, 75syl2anc 691 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) ∈ V)
7732, 52, 54, 56, 76ovmpt2d 6686 . . . . . . . . . 10 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝑂𝑌) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸))
7877oveqd 6566 . . . . . . . . 9 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇))
7978eleq2d 2673 . . . . . . . 8 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) ↔ 𝑊 ∈ (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇)))
80 eqid 2610 . . . . . . . . 9 (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)
8180elmpt2cl 6774 . . . . . . . 8 (𝑊 ∈ (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))
8279, 81syl6bi 242 . . . . . . 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 445 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))
851, 84jca 553 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
8685ex 449 . . 3 ((𝑋𝐴𝑌𝐵) → ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
872mpt2ndm0 6773 . . . . . . 7 (¬ (𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = ∅)
8887oveqd 6566 . . . . . 6 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆𝑇))
8988eleq2d 2673 . . . . 5 (¬ (𝑋𝐴𝑌𝐵) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) ↔ 𝑊 ∈ (𝑆𝑇)))
90 noel 3878 . . . . . . 7 ¬ 𝑊 ∈ ∅
9190pm2.21i 115 . . . . . 6 (𝑊 ∈ ∅ → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
92 0ov 6580 . . . . . 6 (𝑆𝑇) = ∅
9391, 92eleq2s 2706 . . . . 5 (𝑊 ∈ (𝑆𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
9489, 93syl6bi 242 . . . 4 (¬ (𝑋𝐴𝑌𝐵) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
9594adantld 482 . . 3 (¬ (𝑋𝐴𝑌𝐵) → ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
9686, 95pm2.61i 175 . 2 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
9796ex 449 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 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⦋csb 3499  ∅c0 3874  (class class class)co 6549   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060 This theorem is referenced by:  el2mpt2cl  7138
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