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Theorem suppun 7360
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 3809 . . . . . 6 (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
2 cnvun 5573 . . . . . . . 8 (𝐹𝐺) = (𝐹𝐺)
32imaeq1i 5498 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹𝐺) “ (V ∖ {𝑍}))
4 imaundir 5581 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
53, 4eqtri 2673 . . . . . 6 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
61, 5sseqtr4i 3671 . . . . 5 (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹𝐺) “ (V ∖ {𝑍}))
76a1i 11 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹𝐺) “ (V ∖ {𝑍})))
8 suppimacnv 7351 . . . . 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 7001 . . . . . . 7 ((𝐹 ∈ V ∧ 𝐺𝑉) → (𝐹𝐺) ∈ V)
1211adantlr 751 . . . . . 6 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝐺𝑉) → (𝐹𝐺) ∈ V)
1310, 12sylan2 490 . . . . 5 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹𝐺) ∈ V)
14 simplr 807 . . . . 5 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V)
15 suppimacnv 7351 . . . . 5 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
1613, 14, 15syl2anc 694 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
177, 9, 163sstr4d 3681 . . 3 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍))
1817ex 449 . 2 ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍)))
19 supp0prc 7343 . . . 4 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
20 0ss 4005 . . . 4 ∅ ⊆ ((𝐹𝐺) supp 𝑍)
2119, 20syl6eqss 3688 . . 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 176 1 (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cdif 3604  cun 3605  wss 3607  c0 3948  {csn 4210  ccnv 5142  cima 5146  (class class class)co 6690   supp csupp 7340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-supp 7341
This theorem is referenced by:  fsuppunbi  8337  gsumzaddlem  18367
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