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Theorem suppco 7856
 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 7858. (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 7619 . . . . 5 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
2 simpl 486 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → 𝑍 ∈ V)
3 suppimacnv 7827 . . . . 5 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
41, 2, 3syl2an2 685 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
5 cnvco 5721 . . . . . 6 (𝐹𝐺) = (𝐺𝐹)
65imaeq1i 5894 . . . . 5 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
76a1i 11 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍})))
8 imaco 6072 . . . . 5 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
9 simprl 770 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → 𝐹𝑉)
10 suppimacnv 7827 . . . . . . 7 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
119, 2, 10syl2anc 587 . . . . . 6 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
1211imaeq2d 5897 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ (𝐹 “ (V ∖ {𝑍}))))
138, 12eqtr4id 2852 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 supp 𝑍)))
144, 7, 133eqtrd 2837 . . 3 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
1514ex 416 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍))))
16 prcnel 3465 . . . . . 6 𝑍 ∈ V → ¬ 𝑍 ∈ V)
1716intnand 492 . . . . 5 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
18 supp0prc 7819 . . . . 5 (¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ∅)
1917, 18syl 17 . . . 4 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = ∅)
2016intnand 492 . . . . . . 7 𝑍 ∈ V → ¬ (𝐹 ∈ V ∧ 𝑍 ∈ V))
21 supp0prc 7819 . . . . . . 7 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
2220, 21syl 17 . . . . . 6 𝑍 ∈ V → (𝐹 supp 𝑍) = ∅)
2322imaeq2d 5897 . . . . 5 𝑍 ∈ V → (𝐺 “ (𝐹 supp 𝑍)) = (𝐺 “ ∅))
24 ima0 5913 . . . . 5 (𝐺 “ ∅) = ∅
2523, 24eqtrdi 2849 . . . 4 𝑍 ∈ V → (𝐺 “ (𝐹 supp 𝑍)) = ∅)
2619, 25eqtr4d 2836 . . 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 185 1 ((𝐹𝑉𝐺𝑊) → ((𝐹𝐺) supp 𝑍) = (𝐺 “ (𝐹 supp 𝑍)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878  ∅c0 4243  {csn 4525  ◡ccnv 5519   “ cima 5523   ∘ ccom 5524  (class class class)co 7136   supp csupp 7816 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5032  df-opab 5094  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-supp 7817 This theorem is referenced by:  supp0cosupp0  7858  imacosupp  7860
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