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Theorem supp0cosupp0 7501
Description: 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.)
Assertion
Ref Expression
supp0cosupp0 ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))

Proof of Theorem supp0cosupp0
StepHypRef Expression
1 simpl 474 . . . . . . . 8 ((𝐹𝑉𝐺𝑊) → 𝐹𝑉)
21anim2i 594 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ 𝐹𝑉))
32ancomd 466 . . . . . 6 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹𝑉𝑍 ∈ V))
4 suppimacnv 7472 . . . . . 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 7280 . . . . . . . . 9 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
87anim2i 594 . . . . . . . 8 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ (𝐹𝐺) ∈ V))
98ancomd 466 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
10 suppimacnv 7472 . . . . . . 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 5461 . . . . . . . . 9 (𝐹𝐺) = (𝐺𝐹)
1312imaeq1i 5619 . . . . . . . 8 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
14 imaco 5799 . . . . . . . 8 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
1513, 14eqtri 2780 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
16 imaeq2 5618 . . . . . . . 8 ((𝐹 “ (V ∖ {𝑍})) = ∅ → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) = (𝐺 “ ∅))
17 ima0 5637 . . . . . . . 8 (𝐺 “ ∅) = ∅
1816, 17syl6eq 2808 . . . . . . 7 ((𝐹 “ (V ∖ {𝑍})) = ∅ → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) = ∅)
1915, 18syl5eq 2804 . . . . . 6 ((𝐹 “ (V ∖ {𝑍})) = ∅ → ((𝐹𝐺) “ (V ∖ {𝑍})) = ∅)
2011, 19sylan9eq 2812 . . . . 5 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (𝐹 “ (V ∖ {𝑍})) = ∅) → ((𝐹𝐺) supp 𝑍) = ∅)
2120ex 449 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 “ (V ∖ {𝑍})) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
226, 21sylbid 230 . . 3 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
2322ex 449 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅)))
24 id 22 . . . . 5 𝑍 ∈ V → ¬ 𝑍 ∈ V)
2524intnand 1000 . . . 4 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
26 supp0prc 7464 . . . 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 176 1 ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1630  wcel 2137  Vcvv 3338  cdif 3710  c0 4056  {csn 4319  ccnv 5263  cima 5267  ccom 5268  (class class class)co 6811   supp csupp 7461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-rab 3057  df-v 3340  df-sbc 3575  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-op 4326  df-uni 4587  df-br 4803  df-opab 4863  df-id 5172  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-iota 6010  df-fun 6049  df-fv 6055  df-ov 6814  df-oprab 6815  df-mpt2 6816  df-supp 7462
This theorem is referenced by:  gsumval3lem2  18505
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