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Theorem supp0cosupp0 7599
 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 476 . . . . . . . 8 ((𝐹𝑉𝐺𝑊) → 𝐹𝑉)
21anim2i 612 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ 𝐹𝑉))
32ancomd 455 . . . . . 6 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹𝑉𝑍 ∈ V))
4 suppimacnv 7570 . . . . . 6 ((𝐹𝑉𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
53, 4syl 17 . . . . 5 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
65eqeq1d 2827 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) = ∅ ↔ (𝐹 “ (V ∖ {𝑍})) = ∅))
7 coexg 7379 . . . . . . . . 9 ((𝐹𝑉𝐺𝑊) → (𝐹𝐺) ∈ V)
87anim2i 612 . . . . . . . 8 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → (𝑍 ∈ V ∧ (𝐹𝐺) ∈ V))
98ancomd 455 . . . . . . 7 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
10 suppimacnv 7570 . . . . . . 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 5540 . . . . . . . . 9 (𝐹𝐺) = (𝐺𝐹)
1312imaeq1i 5704 . . . . . . . 8 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐺𝐹) “ (V ∖ {𝑍}))
14 imaco 5881 . . . . . . . 8 ((𝐺𝐹) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
1513, 14eqtri 2849 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = (𝐺 “ (𝐹 “ (V ∖ {𝑍})))
16 imaeq2 5703 . . . . . . . 8 ((𝐹 “ (V ∖ {𝑍})) = ∅ → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) = (𝐺 “ ∅))
17 ima0 5722 . . . . . . . 8 (𝐺 “ ∅) = ∅
1816, 17syl6eq 2877 . . . . . . 7 ((𝐹 “ (V ∖ {𝑍})) = ∅ → (𝐺 “ (𝐹 “ (V ∖ {𝑍}))) = ∅)
1915, 18syl5eq 2873 . . . . . 6 ((𝐹 “ (V ∖ {𝑍})) = ∅ → ((𝐹𝐺) “ (V ∖ {𝑍})) = ∅)
2011, 19sylan9eq 2881 . . . . 5 (((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) ∧ (𝐹 “ (V ∖ {𝑍})) = ∅) → ((𝐹𝐺) supp 𝑍) = ∅)
2120ex 403 . . . 4 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 “ (V ∖ {𝑍})) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
226, 21sylbid 232 . . 3 ((𝑍 ∈ V ∧ (𝐹𝑉𝐺𝑊)) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
2322ex 403 . 2 (𝑍 ∈ V → ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅)))
24 id 22 . . . . 5 𝑍 ∈ V → ¬ 𝑍 ∈ V)
2524intnand 484 . . . 4 𝑍 ∈ V → ¬ ((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V))
26 supp0prc 7562 . . . 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 177 1 ((𝐹𝑉𝐺𝑊) → ((𝐹 supp 𝑍) = ∅ → ((𝐹𝐺) supp 𝑍) = ∅))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  Vcvv 3414   ∖ cdif 3795  ∅c0 4144  {csn 4397  ◡ccnv 5341   “ cima 5345   ∘ ccom 5346  (class class class)co 6905   supp csupp 7559 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-supp 7560 This theorem is referenced by:  gsumval3lem2  18660
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