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Theorem supp0cosupp0OLD 7883
 Description: Obsolete version of supp0cosupp0 7882 as of 15-Sep-2023. The support of the composition of two functions is empty if the support of the outer function is empty. (Contributed by AV, 30-May-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
supp0cosupp0OLD ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))

Proof of Theorem supp0cosupp0OLD
StepHypRef Expression
1 simpl 486 . . . . . . . 8 ((𝐹𝑉𝐺𝑊) → 𝐹𝑉)
21anim2i 619 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ 𝐹𝑉))
32ancomd 465 . . . . . 6 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹𝑉𝑍 ∈ V))
4 suppimacnv 7848 . . . . . 6 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
53, 4syl 17 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
65eqeq1d 2760 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹 “ (V ∖ {𝑍})) = ∅))
7 coexg 7639 . . . . . . . . 9 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
87anim2i 619 . . . . . . . 8 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ (𝐹𝐺) ∈ V))
98ancomd 465 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
10 suppimacnv 7848 . . . . . . 7 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
119, 10syl 17 . . . . . 6 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
12 cnvco 5725 . . . . . . . . 9 (𝐹𝐺) = (𝐺𝐹)
1312imaeq1i 5898 . . . . . . . 8 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
14 imaco 6081 . . . . . . . 8 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
1513, 14eqtri 2781 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
16 imaeq2 5897 . . . . . . . 8 ((𝐹 “ (V ∖ {𝑍})) = ∅ → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) = (𝐺 “ ∅))
17 ima0 5917 . . . . . . . 8 (𝐺 “ ∅) = ∅
1816, 17eqtrdi 2809 . . . . . . 7 ((𝐹 “ (V ∖ {𝑍})) = ∅ → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) = ∅)
1915, 18syl5eq 2805 . . . . . 6 ((𝐹 “ (V ∖ {𝑍})) = ∅ → ((𝐹𝐺) “ (V ∖ {𝑍})) = ∅)
2011, 19sylan9eq 2813 . . . . 5 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (𝐹 “ (V ∖ {𝑍})) = ∅) → ((𝐹𝐺) supp 𝑍) = ∅)
2120ex 416 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 “ (V ∖ {𝑍})) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
226, 21sylbid 243 . . 3 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
2322ex 416 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅)))
24 id 22 . . . . 5 𝑍 ∈ V → ¬ 𝑍 ∈ V)
2524intnand 492 . . . 4 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
26 supp0prc 7838 . . . 4 (¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ∅)
2725, 26syl 17 . . 3 𝑍 ∈ V → ((𝐹𝐺) supp 𝑍) = ∅)
28272a1d 26 . 2 𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅)))
2923, 28pm2.61i 185 1 ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ∖ cdif 3855  ∅c0 4225  {csn 4522  ◡ccnv 5523   “ cima 5527   ∘ ccom 5528  (class class class)co 7150   supp csupp 7835 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 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-supp 7836 This theorem is referenced by: (None)
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