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Theorem suppun 8208
Description: The support of a class/function is a subset of the support of the union of this class/function with another class/function. (Contributed by AV, 4-Jun-2019.)
Hypothesis
Ref Expression
suppun.g (𝜑𝐺𝑉)
Assertion
Ref Expression
suppun (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍))

Proof of Theorem suppun
StepHypRef Expression
1 ssun1 4188 . . . . . 6 (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
2 cnvun 6165 . . . . . . . 8 (𝐹𝐺) = (𝐹𝐺)
32imaeq1i 6077 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹𝐺) “ (V ∖ {𝑍}))
4 imaundir 6173 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
53, 4eqtri 2763 . . . . . 6 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
61, 5sseqtrri 4033 . . . . 5 (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹𝐺) “ (V ∖ {𝑍}))
76a1i 11 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹𝐺) “ (V ∖ {𝑍})))
8 suppimacnv 8198 . . . . 5 ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
98adantr 480 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
10 suppun.g . . . . . 6 (𝜑𝐺𝑉)
11 unexg 7762 . . . . . . 7 ((𝐹 ∈ V ∧ 𝐺𝑉) → (𝐹𝐺) ∈ V)
1211adantlr 715 . . . . . 6 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝐺𝑉) → (𝐹𝐺) ∈ V)
1310, 12sylan2 593 . . . . 5 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹𝐺) ∈ V)
14 simplr 769 . . . . 5 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V)
15 suppimacnv 8198 . . . . 5 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
1613, 14, 15syl2anc 584 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
177, 9, 163sstr4d 4043 . . 3 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍))
1817ex 412 . 2 ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍)))
19 supp0prc 8187 . . . 4 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
20 0ss 4406 . . . 4 ∅ ⊆ ((𝐹𝐺) supp 𝑍)
2119, 20eqsstrdi 4050 . . 3 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍))
2221a1d 25 . 2 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍)))
2318, 22pm2.61i 182 1 (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cdif 3960  cun 3961  wss 3963  c0 4339  {csn 4631  ccnv 5688  cima 5692  (class class class)co 7431   supp csupp 8184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8185
This theorem is referenced by:  fsuppunbi  9427  gsumzaddlem  19954
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