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Theorem suppun 7260
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 3754 . . . . . 6 (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
2 cnvun 5497 . . . . . . . 8 (𝐹𝐺) = (𝐹𝐺)
32imaeq1i 5422 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹𝐺) “ (V ∖ {𝑍}))
4 imaundir 5505 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
53, 4eqtri 2643 . . . . . 6 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
61, 5sseqtr4i 3617 . . . . 5 (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹𝐺) “ (V ∖ {𝑍}))
76a1i 11 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹𝐺) “ (V ∖ {𝑍})))
8 suppimacnv 7251 . . . . 5 ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
98adantr 481 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
10 suppun.g . . . . . 6 (𝜑𝐺𝑉)
11 unexg 6912 . . . . . . 7 ((𝐹 ∈ V ∧ 𝐺𝑉) → (𝐹𝐺) ∈ V)
1211adantlr 750 . . . . . 6 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝐺𝑉) → (𝐹𝐺) ∈ V)
1310, 12sylan2 491 . . . . 5 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹𝐺) ∈ V)
14 simplr 791 . . . . 5 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V)
15 suppimacnv 7251 . . . . 5 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
1613, 14, 15syl2anc 692 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
177, 9, 163sstr4d 3627 . . 3 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍))
1817ex 450 . 2 ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍)))
19 supp0prc 7243 . . . 4 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
20 0ss 3944 . . . 4 ∅ ⊆ ((𝐹𝐺) supp 𝑍)
2119, 20syl6eqss 3634 . . 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 384   = wceq 1480  wcel 1987  Vcvv 3186  cdif 3552  cun 3553  wss 3555  c0 3891  {csn 4148  ccnv 5073  cima 5077  (class class class)co 6604   supp csupp 7240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-supp 7241
This theorem is referenced by:  fsuppunbi  8240  gsumzaddlem  18242
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