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Theorem 2mpt20 6780
 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.)
Hypotheses
Ref Expression
2mpt20.o 𝑂 = (𝑥𝐴, 𝑦𝐵𝐸)
2mpt20.u ((𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))
Assertion
Ref Expression
2mpt20 (¬ ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝐶,𝑠,𝑡   𝐷,𝑠,𝑡
Allowed substitution hints:   𝐴(𝑡,𝑠)   𝐵(𝑡,𝑠)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑆(𝑥,𝑦,𝑡,𝑠)   𝑇(𝑥,𝑦,𝑡,𝑠)   𝐸(𝑥,𝑦,𝑡,𝑠)   𝐹(𝑥,𝑦,𝑡,𝑠)   𝑂(𝑥,𝑦,𝑡,𝑠)   𝑋(𝑥,𝑦,𝑡,𝑠)   𝑌(𝑥,𝑦,𝑡,𝑠)

Proof of Theorem 2mpt20
StepHypRef Expression
1 ianor 508 . 2 (¬ ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐶𝑇𝐷)) ↔ (¬ (𝑋𝐴𝑌𝐵) ∨ ¬ (𝑆𝐶𝑇𝐷)))
2 2mpt20.o . . . . . 6 𝑂 = (𝑥𝐴, 𝑦𝐵𝐸)
32mpt2ndm0 6773 . . . . 5 (¬ (𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = ∅)
43oveqd 6566 . . . 4 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆𝑇))
5 0ov 6580 . . . 4 (𝑆𝑇) = ∅
64, 5syl6eq 2660 . . 3 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
7 notnotb 303 . . . 4 ((𝑋𝐴𝑌𝐵) ↔ ¬ ¬ (𝑋𝐴𝑌𝐵))
8 2mpt20.u . . . . . . 7 ((𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))
98adantr 480 . . . . . 6 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))
109oveqd 6566 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇))
11 eqid 2610 . . . . . . 7 (𝑠𝐶, 𝑡𝐷𝐹) = (𝑠𝐶, 𝑡𝐷𝐹)
1211mpt2ndm0 6773 . . . . . 6 (¬ (𝑆𝐶𝑇𝐷) → (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇) = ∅)
1312adantl 481 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇) = ∅)
1410, 13eqtrd 2644 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
157, 14sylanbr 489 . . 3 ((¬ ¬ (𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
166, 15jaoi3 1003 . 2 ((¬ (𝑋𝐴𝑌𝐵) ∨ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
171, 16sylbi 206 1 (¬ ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∅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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  wwlksnon0  41123
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