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Theorem 2mpt20 6924
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 6917 . . . . 5 (¬ (𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = ∅)
43oveqd 6707 . . . 4 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆𝑇))
5 0ov 6722 . . . 4 (𝑆𝑇) = ∅
64, 5syl6eq 2701 . . 3 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
7 notnotb 304 . . . 4 ((𝑋𝐴𝑌𝐵) ↔ ¬ ¬ (𝑋𝐴𝑌𝐵))
8 2mpt20.u . . . . . . 7 ((𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))
98adantr 480 . . . . . 6 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))
109oveqd 6707 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇))
11 eqid 2651 . . . . . . 7 (𝑠𝐶, 𝑡𝐷𝐹) = (𝑠𝐶, 𝑡𝐷𝐹)
1211mpt2ndm0 6917 . . . . . 6 (¬ (𝑆𝐶𝑇𝐷) → (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇) = ∅)
1312adantl 481 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇) = ∅)
1410, 13eqtrd 2685 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
157, 14sylanbr 489 . . 3 ((¬ ¬ (𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
166, 15jaoi3 1031 . 2 ((¬ (𝑋𝐴𝑌𝐵) ∨ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
171, 16sylbi 207 1 (¬ ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  c0 3948  (class class class)co 6690  cmpt2 6692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-dm 5153  df-iota 5889  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695
This theorem is referenced by:  wwlksnon0  26804
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