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Theorem suppco 8187
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 8189. (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 7914 . . . . 5 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
2 simpl 482 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → 𝑍 ∈ V)
3 suppimacnv 8154 . . . . 5 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
41, 2, 3syl2an2 683 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
5 cnvco 5876 . . . . . 6 (𝐹𝐺) = (𝐺𝐹)
65imaeq1i 6047 . . . . 5 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
76a1i 11 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍})))
8 imaco 6241 . . . . 5 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
9 simprl 768 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → 𝐹𝑉)
10 suppimacnv 8154 . . . . . . 7 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
119, 2, 10syl2anc 583 . . . . . 6 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1211imaeq2d 6050 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ (𝐹 “ (V ∖ {𝑍}))))
138, 12eqtr4id 2783 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 supp 𝑍)))
144, 7, 133eqtrd 2768 . . 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 3490 . . . . . 6 𝑍 ∈ V → ¬ 𝑍 ∈ V)
1716intnand 488 . . . . 5 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
18 supp0prc 8144 . . . . 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 8144 . . . . . . 7 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
2220, 21syl 17 . . . . . 6 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
2322imaeq2d 6050 . . . . 5 𝑍 ∈ V → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ ∅))
24 ima0 6067 . . . . 5 (𝐺 “ ∅) = ∅
2523, 24eqtrdi 2780 . . . 4 𝑍 ∈ V → (𝐺 “ (𝐹 supp 𝑍)) = ∅)
2619, 25eqtr4d 2767 . . 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 1533  wcel 2098  Vcvv 3466  cdif 3938  c0 4315  {csn 4621  ccnv 5666  cima 5670  ccom 5671  (class class class)co 7402   supp csupp 8141
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-supp 8142
This theorem is referenced by:  supp0cosupp0  8189  imacosupp  8190
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