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Theorem suppun 7202
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 3738 . . . . . 6 (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
2 cnvun 5457 . . . . . . . 8 (𝐹𝐺) = (𝐹𝐺)
32imaeq1i 5382 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹𝐺) “ (V ∖ {𝑍}))
4 imaundir 5465 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
53, 4eqtri 2632 . . . . . 6 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
61, 5sseqtr4i 3601 . . . . 5 (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹𝐺) “ (V ∖ {𝑍}))
76a1i 11 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹𝐺) “ (V ∖ {𝑍})))
8 suppimacnv 7193 . . . . 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 6857 . . . . . . 7 ((𝐹 ∈ V ∧ 𝐺𝑉) → (𝐹𝐺) ∈ V)
1211adantlr 747 . . . . . 6 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝐺𝑉) → (𝐹𝐺) ∈ V)
1310, 12sylan2 490 . . . . 5 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹𝐺) ∈ V)
14 simplr 788 . . . . 5 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V)
15 suppimacnv 7193 . . . . 5 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
1613, 14, 15syl2anc 691 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
177, 9, 163sstr4d 3611 . . 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 7185 . . . 4 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
20 0ss 3924 . . . 4 ∅ ⊆ ((𝐹𝐺) supp 𝑍)
2119, 20syl6eqss 3618 . . 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 175 1 (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  Vcvv 3173  cdif 3537  cun 3538  wss 3540  c0 3874  {csn 4125  ccnv 5037  cima 5041  (class class class)co 6549   supp csupp 7182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-supp 7183
This theorem is referenced by:  fsuppunbi  8179  gsumzaddlem  18144
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