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Theorem el2mpt2csbcl 7195
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 473 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → (𝑋𝐴𝑌𝐵))
2 el2mpt2csbcl.o . . . . . . . . . . . . 13 𝑂 = (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸))
3 nfcv 2761 . . . . . . . . . . . . . 14 𝑎(𝑠𝐶, 𝑡𝐷𝐸)
4 nfcv 2761 . . . . . . . . . . . . . 14 𝑏(𝑠𝐶, 𝑡𝐷𝐸)
5 nfcsb1v 3530 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐶
6 nfcsb1v 3530 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐷
7 nfcsb1v 3530 . . . . . . . . . . . . . . 15 𝑥𝑎 / 𝑥𝑏 / 𝑦𝐸
85, 6, 7nfmpt2 6677 . . . . . . . . . . . . . 14 𝑥(𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸)
9 nfcv 2761 . . . . . . . . . . . . . . . 16 𝑦𝑎
10 nfcsb1v 3530 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐶
119, 10nfcsb 3532 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐶
12 nfcsb1v 3530 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐷
139, 12nfcsb 3532 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐷
14 nfcsb1v 3530 . . . . . . . . . . . . . . . 16 𝑦𝑏 / 𝑦𝐸
159, 14nfcsb 3532 . . . . . . . . . . . . . . 15 𝑦𝑎 / 𝑥𝑏 / 𝑦𝐸
1611, 13, 15nfmpt2 6677 . . . . . . . . . . . . . 14 𝑦(𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸)
17 csbeq1a 3523 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐶 = 𝑎 / 𝑥𝐶)
18 csbeq1a 3523 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐶 = 𝑏 / 𝑦𝐶)
1918csbeq2dv 3964 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐶 = 𝑎 / 𝑥𝑏 / 𝑦𝐶)
2017, 19sylan9eq 2675 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐶 = 𝑎 / 𝑥𝑏 / 𝑦𝐶)
21 csbeq1a 3523 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐷 = 𝑎 / 𝑥𝐷)
22 csbeq1a 3523 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐷 = 𝑏 / 𝑦𝐷)
2322csbeq2dv 3964 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐷 = 𝑎 / 𝑥𝑏 / 𝑦𝐷)
2421, 23sylan9eq 2675 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐷 = 𝑎 / 𝑥𝑏 / 𝑦𝐷)
25 csbeq1a 3523 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎𝐸 = 𝑎 / 𝑥𝐸)
26 csbeq1a 3523 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑏𝐸 = 𝑏 / 𝑦𝐸)
2726csbeq2dv 3964 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑏𝑎 / 𝑥𝐸 = 𝑎 / 𝑥𝑏 / 𝑦𝐸)
2825, 27sylan9eq 2675 . . . . . . . . . . . . . . 15 ((𝑥 = 𝑎𝑦 = 𝑏) → 𝐸 = 𝑎 / 𝑥𝑏 / 𝑦𝐸)
2920, 24, 28mpt2eq123dv 6670 . . . . . . . . . . . . . 14 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝑠𝐶, 𝑡𝐷𝐸) = (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸))
303, 4, 8, 16, 29cbvmpt2 6687 . . . . . . . . . . . . 13 (𝑥𝐴, 𝑦𝐵 ↦ (𝑠𝐶, 𝑡𝐷𝐸)) = (𝑎𝐴, 𝑏𝐵 ↦ (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸))
312, 30eqtri 2643 . . . . . . . . . . . 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 3517 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑏 / 𝑦𝐶)
3433adantr 481 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑏 / 𝑦𝐶)
35 csbeq1 3517 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐶 = 𝑌 / 𝑦𝐶)
3635adantl 482 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐶 = 𝑌 / 𝑦𝐶)
3736csbeq2dv 3964 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑌 / 𝑦𝐶)
3834, 37eqtrd 2655 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐶 = 𝑋 / 𝑥𝑌 / 𝑦𝐶)
39 csbeq1 3517 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑏 / 𝑦𝐷)
4039adantr 481 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑏 / 𝑦𝐷)
41 csbeq1 3517 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐷 = 𝑌 / 𝑦𝐷)
4241adantl 482 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐷 = 𝑌 / 𝑦𝐷)
4342csbeq2dv 3964 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑌 / 𝑦𝐷)
4440, 43eqtrd 2655 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐷 = 𝑋 / 𝑥𝑌 / 𝑦𝐷)
45 csbeq1 3517 . . . . . . . . . . . . . . 15 (𝑎 = 𝑋𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑏 / 𝑦𝐸)
4645adantr 481 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑏 / 𝑦𝐸)
47 csbeq1 3517 . . . . . . . . . . . . . . . 16 (𝑏 = 𝑌𝑏 / 𝑦𝐸 = 𝑌 / 𝑦𝐸)
4847adantl 482 . . . . . . . . . . . . . . 15 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑏 / 𝑦𝐸 = 𝑌 / 𝑦𝐸)
4948csbeq2dv 3964 . . . . . . . . . . . . . 14 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑋 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑌 / 𝑦𝐸)
5046, 49eqtrd 2655 . . . . . . . . . . . . 13 ((𝑎 = 𝑋𝑏 = 𝑌) → 𝑎 / 𝑥𝑏 / 𝑦𝐸 = 𝑋 / 𝑥𝑌 / 𝑦𝐸)
5138, 44, 50mpt2eq123dv 6670 . . . . . . . . . . . 12 ((𝑎 = 𝑋𝑏 = 𝑌) → (𝑠𝑎 / 𝑥𝑏 / 𝑦𝐶, 𝑡𝑎 / 𝑥𝑏 / 𝑦𝐷𝑎 / 𝑥𝑏 / 𝑦𝐸) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸))
5251adantl 482 . . . . . . . . . . 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 473 . . . . . . . . . . . 12 ((𝑋𝐴𝑌𝐵) → 𝑋𝐴)
5453adantl 482 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋𝐴)
55 simpr 477 . . . . . . . . . . . 12 ((𝑋𝐴𝑌𝐵) → 𝑌𝐵)
5655adantl 482 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑌𝐵)
57 simpl 473 . . . . . . . . . . . . . . . . . 18 ((𝐶𝑈𝐷𝑉) → 𝐶𝑈)
5857ralimi 2947 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑦𝐵 𝐶𝑈)
59 rspcsbela 3978 . . . . . . . . . . . . . . . . 17 ((𝑌𝐵 ∧ ∀𝑦𝐵 𝐶𝑈) → 𝑌 / 𝑦𝐶𝑈)
6055, 58, 59syl2an 494 . . . . . . . . . . . . . . . 16 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑦𝐵 (𝐶𝑈𝐷𝑉)) → 𝑌 / 𝑦𝐶𝑈)
6160ex 450 . . . . . . . . . . . . . . 15 ((𝑋𝐴𝑌𝐵) → (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → 𝑌 / 𝑦𝐶𝑈))
6261ralimdv 2957 . . . . . . . . . . . . . 14 ((𝑋𝐴𝑌𝐵) → (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈))
6362impcom 446 . . . . . . . . . . . . 13 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈)
64 rspcsbela 3978 . . . . . . . . . . . . 13 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝑌 / 𝑦𝐶𝑈) → 𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈)
6554, 63, 64syl2anc 692 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈)
66 simpr 477 . . . . . . . . . . . . . . . . . 18 ((𝐶𝑈𝐷𝑉) → 𝐷𝑉)
6766ralimi 2947 . . . . . . . . . . . . . . . . 17 (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑦𝐵 𝐷𝑉)
68 rspcsbela 3978 . . . . . . . . . . . . . . . . 17 ((𝑌𝐵 ∧ ∀𝑦𝐵 𝐷𝑉) → 𝑌 / 𝑦𝐷𝑉)
6955, 67, 68syl2an 494 . . . . . . . . . . . . . . . 16 (((𝑋𝐴𝑌𝐵) ∧ ∀𝑦𝐵 (𝐶𝑈𝐷𝑉)) → 𝑌 / 𝑦𝐷𝑉)
7069ex 450 . . . . . . . . . . . . . . 15 ((𝑋𝐴𝑌𝐵) → (∀𝑦𝐵 (𝐶𝑈𝐷𝑉) → 𝑌 / 𝑦𝐷𝑉))
7170ralimdv 2957 . . . . . . . . . . . . . 14 ((𝑋𝐴𝑌𝐵) → (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) → ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉))
7271impcom 446 . . . . . . . . . . . . 13 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉)
73 rspcsbela 3978 . . . . . . . . . . . . 13 ((𝑋𝐴 ∧ ∀𝑥𝐴 𝑌 / 𝑦𝐷𝑉) → 𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉)
7454, 72, 73syl2anc 692 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → 𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉)
75 mpt2exga 7191 . . . . . . . . . . . 12 ((𝑋 / 𝑥𝑌 / 𝑦𝐶𝑈𝑋 / 𝑥𝑌 / 𝑦𝐷𝑉) → (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) ∈ V)
7665, 74, 75syl2anc 692 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) ∈ V)
7732, 52, 54, 56, 76ovmpt2d 6741 . . . . . . . . . 10 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑋𝑂𝑌) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸))
7877oveqd 6621 . . . . . . . . 9 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇))
7978eleq2d 2684 . . . . . . . 8 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ (𝑋𝐴𝑌𝐵)) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) ↔ 𝑊 ∈ (𝑆(𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)𝑇)))
80 eqid 2621 . . . . . . . . 9 (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸) = (𝑠𝑋 / 𝑥𝑌 / 𝑦𝐶, 𝑡𝑋 / 𝑥𝑌 / 𝑦𝐷𝑋 / 𝑥𝑌 / 𝑦𝐸)
8180elmpt2cl 6829 . . . . . . . 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 456 . . . . . 6 ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋𝐴𝑌𝐵) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
8483impcom 446 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))
851, 84jca 554 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ (∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇))) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
8685ex 450 . . 3 ((𝑋𝐴𝑌𝐵) → ((∀𝑥𝐴𝑦𝐵 (𝐶𝑈𝐷𝑉) ∧ 𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇)) → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷))))
872mpt2ndm0 6828 . . . . . . 7 (¬ (𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = ∅)
8887oveqd 6621 . . . . . 6 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆𝑇))
8988eleq2d 2684 . . . . 5 (¬ (𝑋𝐴𝑌𝐵) → (𝑊 ∈ (𝑆(𝑋𝑂𝑌)𝑇) ↔ 𝑊 ∈ (𝑆𝑇)))
90 noel 3895 . . . . . . 7 ¬ 𝑊 ∈ ∅
9190pm2.21i 116 . . . . . 6 (𝑊 ∈ ∅ → ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝑋 / 𝑥𝑌 / 𝑦𝐶𝑇𝑋 / 𝑥𝑌 / 𝑦𝐷)))
92 0ov 6635 . . . . . 6 (𝑆𝑇) = ∅
9391, 92eleq2s 2716 . . . . 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 483 . . 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 450 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 384   = wceq 1480  wcel 1987  wral 2907  Vcvv 3186  csb 3514  c0 3891  (class class class)co 6604  cmpt2 6606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114
This theorem is referenced by:  el2mpt2cl  7196
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