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Theorem suppco 8230
Description: The support of the composition of two functions is the inverse image by the inner function of the support of the outer function. (Contributed by AV, 30-May-2019.) Extract this statement from the proof of supp0cosupp0 8232. (Revised by SN, 15-Sep-2023.)
Assertion
Ref Expression
suppco ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))

Proof of Theorem suppco
StepHypRef Expression
1 coexg 7952 . . . . 5 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
2 simpl 482 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → 𝑍 ∈ V)
3 suppimacnv 8198 . . . . 5 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
41, 2, 3syl2an2 686 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
5 cnvco 5899 . . . . . 6 (𝐹𝐺) = (𝐺𝐹)
65imaeq1i 6077 . . . . 5 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
76a1i 11 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍})))
8 imaco 6273 . . . . 5 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
9 simprl 771 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → 𝐹𝑉)
10 suppimacnv 8198 . . . . . . 7 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
119, 2, 10syl2anc 584 . . . . . 6 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1211imaeq2d 6080 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ (𝐹 “ (V ∖ {𝑍}))))
138, 12eqtr4id 2794 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 supp 𝑍)))
144, 7, 133eqtrd 2779 . . 3 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
1514ex 412 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍))))
16 prcnel 3505 . . . . . 6 𝑍 ∈ V → ¬ 𝑍 ∈ V)
1716intnand 488 . . . . 5 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
18 supp0prc 8187 . . . . 5 (¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ∅)
1917, 18syl 17 . . . 4 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = ∅)
2016intnand 488 . . . . . . 7 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
21 supp0prc 8187 . . . . . . 7 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
2220, 21syl 17 . . . . . 6 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
2322imaeq2d 6080 . . . . 5 𝑍 ∈ V → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ ∅))
24 ima0 6097 . . . . 5 (𝐺 “ ∅) = ∅
2523, 24eqtrdi 2791 . . . 4 𝑍 ∈ V → (𝐺 “ (𝐹 supp 𝑍)) = ∅)
2619, 25eqtr4d 2778 . . 3 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
2726a1d 25 . 2 𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍))))
2815, 27pm2.61i 182 1 ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  c0 4339  {csn 4631  ccnv 5688  cima 5692  ccom 5693  (class class class)co 7431   supp csupp 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8185
This theorem is referenced by:  supp0cosupp0  8232  imacosupp  8233
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