MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  suppun Structured version   Visualization version   GIF version

Theorem suppun 8156
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 4170 . . . . . 6 (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
2 cnvun 6134 . . . . . . . 8 (𝐹𝐺) = (𝐹𝐺)
32imaeq1i 6049 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹𝐺) “ (V ∖ {𝑍}))
4 imaundir 6142 . . . . . . 7 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
53, 4eqtri 2761 . . . . . 6 ((𝐹𝐺) “ (V ∖ {𝑍})) = ((𝐹 “ (V ∖ {𝑍})) ∪ (𝐺 “ (V ∖ {𝑍})))
61, 5sseqtrri 4017 . . . . 5 (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹𝐺) “ (V ∖ {𝑍}))
76a1i 11 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 “ (V ∖ {𝑍})) ⊆ ((𝐹𝐺) “ (V ∖ {𝑍})))
8 suppimacnv 8146 . . . . 5 ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
98adantr 482 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) = (𝐹 “ (V ∖ {𝑍})))
10 suppun.g . . . . . 6 (𝜑𝐺𝑉)
11 unexg 7723 . . . . . . 7 ((𝐹 ∈ V ∧ 𝐺𝑉) → (𝐹𝐺) ∈ V)
1211adantlr 714 . . . . . 6 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝐺𝑉) → (𝐹𝐺) ∈ V)
1310, 12sylan2 594 . . . . 5 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹𝐺) ∈ V)
14 simplr 768 . . . . 5 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → 𝑍 ∈ V)
15 suppimacnv 8146 . . . . 5 (((𝐹𝐺) ∈ V ∧ 𝑍 ∈ V) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
1613, 14, 15syl2anc 585 . . . 4 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → ((𝐹𝐺) supp 𝑍) = ((𝐹𝐺) “ (V ∖ {𝑍})))
177, 9, 163sstr4d 4027 . . 3 (((𝐹 ∈ V ∧ 𝑍 ∈ V) ∧ 𝜑) → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍))
1817ex 414 . 2 ((𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝜑 → (𝐹 supp 𝑍) ⊆ ((𝐹𝐺) supp 𝑍)))
19 supp0prc 8136 . . . 4 (¬ (𝐹 ∈ V ∧ 𝑍 ∈ V) → (𝐹 supp 𝑍) = ∅)
20 0ss 4394 . . . 4 ∅ ⊆ ((𝐹𝐺) supp 𝑍)
2119, 20eqsstrdi 4034 . . 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 397   = wceq 1542  wcel 2107  Vcvv 3475  cdif 3943  cun 3944  wss 3946  c0 4320  {csn 4624  ccnv 5671  cima 5675  (class class class)co 7396   supp csupp 8133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-supp 8134
This theorem is referenced by:  fsuppunbi  9372  gsumzaddlem  19772
  Copyright terms: Public domain W3C validator