MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  el2mpt2csbcl Structured version   Visualization version   GIF version

Theorem el2mpt2csbcl 7420
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 474 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → (𝑋𝐴𝑌𝐵))
2 el2mpt2csbcl.o . . . . . . . . . . . . 13 𝑂 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))
3 nfcv 2903 . . . . . . . . . . . . . 14 𝑎(𝑠𝐶, 𝑡𝐷𝐸)
4 nfcv 2903 . . . . . . . . . . . . . 14 𝑏(𝑠𝐶, 𝑡𝐷𝐸)
5 nfcsb1v 3691 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐶
6 nfcsb1v 3691 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐷
7 nfcsb1v 3691 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐸
85, 6, 7nfmpt2 6891 . . . . . . . . . . . . . 14 𝑥(𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸)
9 nfcv 2903 . . . . . . . . . . . . . . . 16 𝑦𝑎
10 nfcsb1v 3691 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐶
119, 10nfcsb 3693 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐶
12 nfcsb1v 3691 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐷
139, 12nfcsb 3693 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐷
14 nfcsb1v 3691 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐸
159, 14nfcsb 3693 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐸
1611, 13, 15nfmpt2 6891 . . . . . . . . . . . . . 14 𝑦(𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸)
17 csbeq1a 3684 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
18 csbeq1a 3684 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐶 = 𝑏 / 𝑦𝐶)
1918csbeq2dv 4136 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐶 = 𝑎 / 𝑥𝑏 / 𝑦𝐶)
2017, 19sylan9eq 2815 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐶 = 𝑎 / 𝑥𝑏 / 𝑦𝐶)
21 csbeq1a 3684 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐷 = 𝑎 / 𝑥𝐷)
22 csbeq1a 3684 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐷 = 𝑏 / 𝑦𝐷)
2322csbeq2dv 4136 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐷 = 𝑎 / 𝑥𝑏 / 𝑦𝐷)
2421, 23sylan9eq 2815 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐷 = 𝑎 / 𝑥𝑏 / 𝑦𝐷)
25 csbeq1a 3684 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐸 = 𝑎 / 𝑥𝐸)
26 csbeq1a 3684 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐸 = 𝑏 / 𝑦𝐸)
2726csbeq2dv 4136 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐸 = 𝑎 / 𝑥𝑏 / 𝑦𝐸)
2825, 27sylan9eq 2815 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐸 = 𝑎 / 𝑥𝑏 / 𝑦𝐸)
2920, 24, 28mpt2eq123dv 6884 . . . . . . . . . . . . . 14 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑠𝐶, 𝑡𝐷𝐸) = (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸))
303, 4, 8, 16, 29cbvmpt2 6901 . . . . . . . . . . . . 13 (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸)) = (𝑎𝐴, 𝑏𝐵 ↦ (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸))
312, 30eqtri 2783 . . . . . . . . . . . 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 3678 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑏 / 𝑦𝐶)
3433adantr 472 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑏 / 𝑦𝐶)
35 csbeq1 3678 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐶 = 𝑌 / 𝑦𝐶)
3635adantl 473 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐶 = 𝑌 / 𝑦𝐶)
3736csbeq2dv 4136 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑌 / 𝑦𝐶)
3834, 37eqtrd 2795 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑌 / 𝑦𝐶)
39 csbeq1 3678 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑏 / 𝑦𝐷)
4039adantr 472 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑏 / 𝑦𝐷)
41 csbeq1 3678 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐷 = 𝑌 / 𝑦𝐷)
4241adantl 473 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐷 = 𝑌 / 𝑦𝐷)
4342csbeq2dv 4136 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑌 / 𝑦𝐷)
4440, 43eqtrd 2795 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑌 / 𝑦𝐷)
45 csbeq1 3678 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑏 / 𝑦𝐸)
4645adantr 472 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑏 / 𝑦𝐸)
47 csbeq1 3678 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐸 = 𝑌 / 𝑦𝐸)
4847adantl 473 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐸 = 𝑌 / 𝑦𝐸)
4948csbeq2dv 4136 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑌 / 𝑦𝐸)
5046, 49eqtrd 2795 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑌 / 𝑦𝐸)
5138, 44, 50mpt2eq123dv 6884 . . . . . . . . . . . 12 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸))
5251adantl 473 . . . . . . . . . . 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 474 . . . . . . . . . . . 12 ((𝑋𝐴𝑌𝐵) → 𝑋𝐴)
5453adantl 473 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋𝐴)
55 simpr 479 . . . . . . . . . . . 12 ((𝑋𝐴𝑌𝐵) → 𝑌𝐵)
5655adantl 473 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑌𝐵)
57 simpl 474 . . . . . . . . . . . . . . . . . 18 ((𝐶𝑈𝐷𝑉) → 𝐶𝑈)
5857ralimi 3091 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑦𝐵 𝐶𝑈)
59 rspcsbela 4150 . . . . . . . . . . . . . . . . 17 ((𝑌𝐵 ∧ ∀𝑦𝐵 𝐶𝑈) → 𝑌 / 𝑦𝐶𝑈)
6055, 58, 59syl2an 495 . . . . . . . . . . . . . . . 16 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑦𝐵 (𝐶𝑈𝐷𝑉)) → 𝑌 / 𝑦𝐶𝑈)
6160ex 449 . . . . . . . . . . . . . . 15 ((𝑋𝐴𝑌𝐵) → (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → 𝑌 / 𝑦𝐶𝑈))
6261ralimdv 3102 . . . . . . . . . . . . . 14 ((𝑋𝐴𝑌𝐵) → (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈))
6362impcom 445 . . . . . . . . . . . . 13 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈)
64 rspcsbela 4150 . . . . . . . . . . . . 13 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈) → 𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈)
6554, 63, 64syl2anc 696 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈)
66 simpr 479 . . . . . . . . . . . . . . . . . 18 ((𝐶𝑈𝐷𝑉) → 𝐷𝑉)
6766ralimi 3091 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑦𝐵 𝐷𝑉)
68 rspcsbela 4150 . . . . . . . . . . . . . . . . 17 ((𝑌𝐵 ∧ ∀𝑦𝐵 𝐷𝑉) → 𝑌 / 𝑦𝐷𝑉)
6955, 67, 68syl2an 495 . . . . . . . . . . . . . . . 16 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑦𝐵 (𝐶𝑈𝐷𝑉)) → 𝑌 / 𝑦𝐷𝑉)
7069ex 449 . . . . . . . . . . . . . . 15 ((𝑋𝐴𝑌𝐵) → (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → 𝑌 / 𝑦𝐷𝑉))
7170ralimdv 3102 . . . . . . . . . . . . . 14 ((𝑋𝐴𝑌𝐵) → (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉))
7271impcom 445 . . . . . . . . . . . . 13 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉)
73 rspcsbela 4150 . . . . . . . . . . . . 13 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉) → 𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉)
7454, 72, 73syl2anc 696 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉)
75 mpt2exga 7416 . . . . . . . . . . . 12 ((𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉) → (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) ∈ V)
7665, 74, 75syl2anc 696 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) ∈ V)
7732, 52, 54, 56, 76ovmpt2d 6955 . . . . . . . . . 10 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝑂𝑌) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸))
7877oveqd 6832 . . . . . . . . 9 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇))
7978eleq2d 2826 . . . . . . . 8 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) ↔ 𝑊 ∈ (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇)))
80 eqid 2761 . . . . . . . . 9 (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)
8180elmpt2cl 7043 . . . . . . . 8 (𝑊 ∈ (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))
8279, 81syl6bi 243 . . . . . . 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 555 . . . 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 7042 . . . . . . 7 (¬ (𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = ∅)
8887oveqd 6832 . . . . . 6 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆𝑇))
8988eleq2d 2826 . . . . 5 (¬ (𝑋𝐴𝑌𝐵) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) ↔ 𝑊 ∈ (𝑆𝑇)))
90 noel 4063 . . . . . . 7 ¬ 𝑊 ∈ ∅
9190pm2.21i 116 . . . . . 6 (𝑊 ∈ ∅ → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
92 0ov 6847 . . . . . 6 (𝑆𝑇) = ∅
9391, 92eleq2s 2858 . . . . 5 (𝑊 ∈ (𝑆𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
9489, 93syl6bi 243 . . . 4 (¬ (𝑋𝐴𝑌𝐵) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
9594adantld 484 . . 3 (¬ (𝑋𝐴𝑌𝐵) → ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
9686, 95pm2.61i 176 . 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 1632  wcel 2140  wral 3051  Vcvv 3341  csb 3675  c0 4059  (class class class)co 6815  cmpt2 6817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-1st 7335  df-2nd 7336
This theorem is referenced by:  el2mpt2cl  7421
  Copyright terms: Public domain W3C validator