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Theorem 2mpt20 7116
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 1005 . 2 (¬ ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐶𝑇𝐷)) ↔ (¬ (𝑋𝐴𝑌𝐵) ∨ ¬ (𝑆𝐶𝑇𝐷)))
2 2mpt20.o . . . . . 6 𝑂 = (𝑥𝐴, 𝑦𝐵𝐸)
32mpt2ndm0 7109 . . . . 5 (¬ (𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = ∅)
43oveqd 6895 . . . 4 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆𝑇))
5 0ov 6914 . . . 4 (𝑆𝑇) = ∅
64, 5syl6eq 2849 . . 3 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
7 notnotb 307 . . . 4 ((𝑋𝐴𝑌𝐵) ↔ ¬ ¬ (𝑋𝐴𝑌𝐵))
8 2mpt20.u . . . . . . 7 ((𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))
98adantr 473 . . . . . 6 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))
109oveqd 6895 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇))
11 eqid 2799 . . . . . . 7 (𝑠𝐶, 𝑡𝐷𝐹) = (𝑠𝐶, 𝑡𝐷𝐹)
1211mpt2ndm0 7109 . . . . . 6 (¬ (𝑆𝐶𝑇𝐷) → (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇) = ∅)
1312adantl 474 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇) = ∅)
1410, 13eqtrd 2833 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
157, 14sylanbr 578 . . 3 ((¬ ¬ (𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
166, 15jaoi3 1084 . 2 ((¬ (𝑋𝐴𝑌𝐵) ∨ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
171, 16sylbi 209 1 (¬ ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 385  wo 874   = wceq 1653  wcel 2157  c0 4115  (class class class)co 6878  cmpt2 6880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-xp 5318  df-dm 5322  df-iota 6064  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883
This theorem is referenced by:  wwlksnon0  27107
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