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Theorem 2mpo0 7388
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
2mpo0.o 𝑂 = (𝑥𝐴, 𝑦𝐵𝐸)
2mpo0.u ((𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))
Assertion
Ref Expression
2mpo0 (¬ ((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 2mpo0
StepHypRef Expression
1 ianor 977 . 2 (¬ ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐶𝑇𝐷)) ↔ (¬ (𝑋𝐴𝑌𝐵) ∨ ¬ (𝑆𝐶𝑇𝐷)))
2 2mpo0.o . . . . . 6 𝑂 = (𝑥𝐴, 𝑦𝐵𝐸)
32mpondm0 7380 . . . . 5 (¬ (𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = ∅)
43oveqd 7167 . . . 4 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆𝑇))
5 0ov 7187 . . . 4 (𝑆𝑇) = ∅
64, 5syl6eq 2877 . . 3 (¬ (𝑋𝐴𝑌𝐵) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
7 notnotb 316 . . . 4 ((𝑋𝐴𝑌𝐵) ↔ ¬ ¬ (𝑋𝐴𝑌𝐵))
8 2mpo0.u . . . . . . 7 ((𝑋𝐴𝑌𝐵) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))
98adantr 481 . . . . . 6 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑋𝑂𝑌) = (𝑠𝐶, 𝑡𝐷𝐹))
109oveqd 7167 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇))
11 eqid 2826 . . . . . . 7 (𝑠𝐶, 𝑡𝐷𝐹) = (𝑠𝐶, 𝑡𝐷𝐹)
1211mpondm0 7380 . . . . . 6 (¬ (𝑆𝐶𝑇𝐷) → (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇) = ∅)
1312adantl 482 . . . . 5 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑠𝐶, 𝑡𝐷𝐹)𝑇) = ∅)
1410, 13eqtrd 2861 . . . 4 (((𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
157, 14sylanbr 582 . . 3 ((¬ ¬ (𝑋𝐴𝑌𝐵) ∧ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
166, 15jaoi3 1054 . 2 ((¬ (𝑋𝐴𝑌𝐵) ∨ ¬ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
171, 16sylbi 218 1 (¬ ((𝑋𝐴𝑌𝐵) ∧ (𝑆𝐶𝑇𝐷)) → (𝑆(𝑋𝑂𝑌)𝑇) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 843   = wceq 1530  wcel 2107  c0 4295  (class class class)co 7150  cmpo 7152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-xp 5560  df-dm 5564  df-iota 6313  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155
This theorem is referenced by:  wwlksnon0  27565
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