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Theorem suppofssd 8188
Description: Condition for the support of a function operation to be a subset of the union of the supports of the left and right function terms. (Contributed by Steven Nguyen, 28-Aug-2023.)
Hypotheses
Ref Expression
suppofssd.1 (𝜑𝐴𝑉)
suppofssd.2 (𝜑𝑍𝐵)
suppofssd.3 (𝜑𝐹:𝐴𝐵)
suppofssd.4 (𝜑𝐺:𝐴𝐵)
suppofssd.5 (𝜑 → (𝑍𝑋𝑍) = 𝑍)
Assertion
Ref Expression
suppofssd (𝜑 → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))

Proof of Theorem suppofssd
Dummy variables 𝑥 𝑦 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovexd 7444 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝑋𝑦) ∈ V)
2 suppofssd.3 . . 3 (𝜑𝐹:𝐴𝐵)
3 suppofssd.4 . . 3 (𝜑𝐺:𝐴𝐵)
4 suppofssd.1 . . 3 (𝜑𝐴𝑉)
5 inidm 4219 . . 3 (𝐴𝐴) = 𝐴
61, 2, 3, 4, 4, 5off 7688 . 2 (𝜑 → (𝐹f 𝑋𝐺):𝐴⟶V)
7 eldif 3959 . . . 4 (𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘𝐴 ∧ ¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))))
8 ioran 983 . . . . . 6 (¬ (𝑘 ∈ (𝐹 supp 𝑍) ∨ 𝑘 ∈ (𝐺 supp 𝑍)) ↔ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))
9 elun 4149 . . . . . 6 (𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ↔ (𝑘 ∈ (𝐹 supp 𝑍) ∨ 𝑘 ∈ (𝐺 supp 𝑍)))
108, 9xchnxbir 333 . . . . 5 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ↔ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))
1110anbi2i 624 . . . 4 ((𝑘𝐴 ∧ ¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))))
127, 11bitri 275 . . 3 (𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))))
132ffnd 6719 . . . . . . . . . 10 (𝜑𝐹 Fn 𝐴)
14 suppofssd.2 . . . . . . . . . 10 (𝜑𝑍𝐵)
15 elsuppfn 8156 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝐴𝑉𝑍𝐵) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍)))
1613, 4, 14, 15syl3anc 1372 . . . . . . . . 9 (𝜑 → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍)))
1716notbid 318 . . . . . . . 8 (𝜑 → (¬ 𝑘 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍)))
1817biimpd 228 . . . . . . 7 (𝜑 → (¬ 𝑘 ∈ (𝐹 supp 𝑍) → ¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍)))
193ffnd 6719 . . . . . . . . . 10 (𝜑𝐺 Fn 𝐴)
20 elsuppfn 8156 . . . . . . . . . 10 ((𝐺 Fn 𝐴𝐴𝑉𝑍𝐵) → (𝑘 ∈ (𝐺 supp 𝑍) ↔ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))
2119, 4, 14, 20syl3anc 1372 . . . . . . . . 9 (𝜑 → (𝑘 ∈ (𝐺 supp 𝑍) ↔ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))
2221notbid 318 . . . . . . . 8 (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑍) ↔ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))
2322biimpd 228 . . . . . . 7 (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑍) → ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))
2418, 23anim12d 610 . . . . . 6 (𝜑 → ((¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)) → (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍))))
2524anim2d 613 . . . . 5 (𝜑 → ((𝑘𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))) → (𝑘𝐴 ∧ (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))))
2625imp 408 . . . 4 ((𝜑 ∧ (𝑘𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))) → (𝑘𝐴 ∧ (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍))))
27 pm3.2 471 . . . . . . . 8 (𝑘𝐴 → ((𝐹𝑘) ≠ 𝑍 → (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍)))
2827necon1bd 2959 . . . . . . 7 (𝑘𝐴 → (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) → (𝐹𝑘) = 𝑍))
29 pm3.2 471 . . . . . . . 8 (𝑘𝐴 → ((𝐺𝑘) ≠ 𝑍 → (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))
3029necon1bd 2959 . . . . . . 7 (𝑘𝐴 → (¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍) → (𝐺𝑘) = 𝑍))
3128, 30anim12d 610 . . . . . 6 (𝑘𝐴 → ((¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)) → ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍)))
3231imdistani 570 . . . . 5 ((𝑘𝐴 ∧ (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍))) → (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍)))
3313adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → 𝐹 Fn 𝐴)
3419adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → 𝐺 Fn 𝐴)
354adantr 482 . . . . . . 7 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → 𝐴𝑉)
36 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → 𝑘𝐴)
37 fnfvof 7687 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝑉𝑘𝐴)) → ((𝐹f 𝑋𝐺)‘𝑘) = ((𝐹𝑘)𝑋(𝐺𝑘)))
3833, 34, 35, 36, 37syl22anc 838 . . . . . 6 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → ((𝐹f 𝑋𝐺)‘𝑘) = ((𝐹𝑘)𝑋(𝐺𝑘)))
39 oveq12 7418 . . . . . . 7 (((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍) → ((𝐹𝑘)𝑋(𝐺𝑘)) = (𝑍𝑋𝑍))
4039ad2antll 728 . . . . . 6 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → ((𝐹𝑘)𝑋(𝐺𝑘)) = (𝑍𝑋𝑍))
41 suppofssd.5 . . . . . . 7 (𝜑 → (𝑍𝑋𝑍) = 𝑍)
4241adantr 482 . . . . . 6 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → (𝑍𝑋𝑍) = 𝑍)
4338, 40, 423eqtrd 2777 . . . . 5 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → ((𝐹f 𝑋𝐺)‘𝑘) = 𝑍)
4432, 43sylan2 594 . . . 4 ((𝜑 ∧ (𝑘𝐴 ∧ (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))) → ((𝐹f 𝑋𝐺)‘𝑘) = 𝑍)
4526, 44syldan 592 . . 3 ((𝜑 ∧ (𝑘𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))) → ((𝐹f 𝑋𝐺)‘𝑘) = 𝑍)
4612, 45sylan2b 595 . 2 ((𝜑𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))) → ((𝐹f 𝑋𝐺)‘𝑘) = 𝑍)
476, 46suppss 8179 1 (𝜑 → ((𝐹f 𝑋𝐺) supp 𝑍) ⊆ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  wne 2941  Vcvv 3475  cdif 3946  cun 3947  wss 3949   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  f cof 7668   supp csupp 8146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-of 7670  df-supp 8147
This theorem is referenced by:  psrbagaddcl  21481  mhpmulcl  21692  mhpaddcl  21694  naddcnff  42112
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