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Theorem suppofssd 7854
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 7174 . . 3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥𝑋𝑦) ∈ V)
2 suppofssd.3 . . 3 (𝜑𝐹:𝐴𝐵)
3 suppofssd.4 . . 3 (𝜑𝐺:𝐴𝐵)
4 suppofssd.1 . . 3 (𝜑𝐴𝑉)
5 inidm 4148 . . 3 (𝐴𝐴) = 𝐴
61, 2, 3, 4, 4, 5off 7408 . 2 (𝜑 → (𝐹f 𝑋𝐺):𝐴⟶V)
7 eldif 3894 . . . 4 (𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘𝐴 ∧ ¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))))
8 ioran 981 . . . . . 6 (¬ (𝑘 ∈ (𝐹 supp 𝑍) ∨ 𝑘 ∈ (𝐺 supp 𝑍)) ↔ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))
9 elun 4079 . . . . . 6 (𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ↔ (𝑘 ∈ (𝐹 supp 𝑍) ∨ 𝑘 ∈ (𝐺 supp 𝑍)))
108, 9xchnxbir 336 . . . . 5 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)) ↔ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))
1110anbi2i 625 . . . 4 ((𝑘𝐴 ∧ ¬ 𝑘 ∈ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))))
127, 11bitri 278 . . 3 (𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍))) ↔ (𝑘𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))))
132ffnd 6492 . . . . . . . . . 10 (𝜑𝐹 Fn 𝐴)
14 suppofssd.2 . . . . . . . . . 10 (𝜑𝑍𝐵)
15 elsuppfn 7825 . . . . . . . . . 10 ((𝐹 Fn 𝐴𝐴𝑉𝑍𝐵) → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍)))
1613, 4, 14, 15syl3anc 1368 . . . . . . . . 9 (𝜑 → (𝑘 ∈ (𝐹 supp 𝑍) ↔ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍)))
1716notbid 321 . . . . . . . 8 (𝜑 → (¬ 𝑘 ∈ (𝐹 supp 𝑍) ↔ ¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍)))
1817biimpd 232 . . . . . . 7 (𝜑 → (¬ 𝑘 ∈ (𝐹 supp 𝑍) → ¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍)))
193ffnd 6492 . . . . . . . . . 10 (𝜑𝐺 Fn 𝐴)
20 elsuppfn 7825 . . . . . . . . . 10 ((𝐺 Fn 𝐴𝐴𝑉𝑍𝐵) → (𝑘 ∈ (𝐺 supp 𝑍) ↔ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))
2119, 4, 14, 20syl3anc 1368 . . . . . . . . 9 (𝜑 → (𝑘 ∈ (𝐺 supp 𝑍) ↔ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))
2221notbid 321 . . . . . . . 8 (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑍) ↔ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))
2322biimpd 232 . . . . . . 7 (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑍) → ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))
2418, 23anim12d 611 . . . . . 6 (𝜑 → ((¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)) → (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍))))
2524anim2d 614 . . . . 5 (𝜑 → ((𝑘𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍))) → (𝑘𝐴 ∧ (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))))
2625imp 410 . . . 4 ((𝜑 ∧ (𝑘𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))) → (𝑘𝐴 ∧ (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍))))
27 pm3.2 473 . . . . . . . 8 (𝑘𝐴 → ((𝐹𝑘) ≠ 𝑍 → (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍)))
2827necon1bd 3008 . . . . . . 7 (𝑘𝐴 → (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) → (𝐹𝑘) = 𝑍))
29 pm3.2 473 . . . . . . . 8 (𝑘𝐴 → ((𝐺𝑘) ≠ 𝑍 → (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))
3029necon1bd 3008 . . . . . . 7 (𝑘𝐴 → (¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍) → (𝐺𝑘) = 𝑍))
3128, 30anim12d 611 . . . . . 6 (𝑘𝐴 → ((¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)) → ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍)))
3231imdistani 572 . . . . 5 ((𝑘𝐴 ∧ (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍))) → (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍)))
3313adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → 𝐹 Fn 𝐴)
3419adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → 𝐺 Fn 𝐴)
354adantr 484 . . . . . . 7 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → 𝐴𝑉)
36 simprl 770 . . . . . . 7 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → 𝑘𝐴)
37 fnfvof 7407 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (𝐴𝑉𝑘𝐴)) → ((𝐹f 𝑋𝐺)‘𝑘) = ((𝐹𝑘)𝑋(𝐺𝑘)))
3833, 34, 35, 36, 37syl22anc 837 . . . . . 6 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → ((𝐹f 𝑋𝐺)‘𝑘) = ((𝐹𝑘)𝑋(𝐺𝑘)))
39 oveq12 7148 . . . . . . 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 484 . . . . . 6 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → (𝑍𝑋𝑍) = 𝑍)
4338, 40, 423eqtrd 2840 . . . . 5 ((𝜑 ∧ (𝑘𝐴 ∧ ((𝐹𝑘) = 𝑍 ∧ (𝐺𝑘) = 𝑍))) → ((𝐹f 𝑋𝐺)‘𝑘) = 𝑍)
4432, 43sylan2 595 . . . 4 ((𝜑 ∧ (𝑘𝐴 ∧ (¬ (𝑘𝐴 ∧ (𝐹𝑘) ≠ 𝑍) ∧ ¬ (𝑘𝐴 ∧ (𝐺𝑘) ≠ 𝑍)))) → ((𝐹f 𝑋𝐺)‘𝑘) = 𝑍)
4526, 44syldan 594 . . 3 ((𝜑 ∧ (𝑘𝐴 ∧ (¬ 𝑘 ∈ (𝐹 supp 𝑍) ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑍)))) → ((𝐹f 𝑋𝐺)‘𝑘) = 𝑍)
4612, 45sylan2b 596 . 2 ((𝜑𝑘 ∈ (𝐴 ∖ ((𝐹 supp 𝑍) ∪ (𝐺 supp 𝑍)))) → ((𝐹f 𝑋𝐺)‘𝑘) = 𝑍)
476, 46suppss 7847 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 209  wa 399  wo 844   = wceq 1538  wcel 2112  wne 2990  Vcvv 3444  cdif 3881  cun 3882  wss 3884   Fn wfn 6323  wf 6324  cfv 6328  (class class class)co 7139  f cof 7391   supp csupp 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-of 7393  df-supp 7818
This theorem is referenced by:  mhpaddcl  20802
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