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

Theorem suppcoss 8230
Description: The support of the composition of two functions is a subset of the support of the inner function if the outer function preserves zero. Compare suppssfv 8225, which has a sethood condition on 𝐴 instead of 𝐵. (Contributed by SN, 25-May-2024.)
Hypotheses
Ref Expression
suppcoss.f (𝜑𝐹 Fn 𝐴)
suppcoss.g (𝜑𝐺:𝐵𝐴)
suppcoss.b (𝜑𝐵𝑊)
suppcoss.y (𝜑𝑌𝑉)
suppcoss.1 (𝜑 → (𝐹𝑌) = 𝑍)
Assertion
Ref Expression
suppcoss (𝜑 → ((𝐹𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))

Proof of Theorem suppcoss
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 suppcoss.f . . . 4 (𝜑𝐹 Fn 𝐴)
2 dffn3 6748 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
31, 2sylib 218 . . 3 (𝜑𝐹:𝐴⟶ran 𝐹)
4 suppcoss.g . . 3 (𝜑𝐺:𝐵𝐴)
53, 4fcod 6761 . 2 (𝜑 → (𝐹𝐺):𝐵⟶ran 𝐹)
6 eldif 3972 . . . . 5 (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)))
74ffnd 6737 . . . . . . . . 9 (𝜑𝐺 Fn 𝐵)
8 suppcoss.b . . . . . . . . 9 (𝜑𝐵𝑊)
9 suppcoss.y . . . . . . . . 9 (𝜑𝑌𝑉)
10 elsuppfn 8193 . . . . . . . . 9 ((𝐺 Fn 𝐵𝐵𝑊𝑌𝑉) → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
117, 8, 9, 10syl3anc 1370 . . . . . . . 8 (𝜑 → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
1211notbid 318 . . . . . . 7 (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑌) ↔ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
1312anbi2d 630 . . . . . 6 (𝜑 → ((𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌))))
14 annotanannot 835 . . . . . 6 ((𝑘𝐵 ∧ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌))
1513, 14bitrdi 287 . . . . 5 (𝜑 → ((𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌)))
166, 15bitrid 283 . . . 4 (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌)))
17 nne 2941 . . . . . 6 (¬ (𝐺𝑘) ≠ 𝑌 ↔ (𝐺𝑘) = 𝑌)
1817anbi2i 623 . . . . 5 ((𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌))
194adantr 480 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → 𝐺:𝐵𝐴)
20 simprl 771 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → 𝑘𝐵)
2119, 20fvco3d 7008 . . . . . . 7 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → ((𝐹𝐺)‘𝑘) = (𝐹‘(𝐺𝑘)))
22 simprr 773 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → (𝐺𝑘) = 𝑌)
2322fveq2d 6910 . . . . . . 7 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → (𝐹‘(𝐺𝑘)) = (𝐹𝑌))
24 suppcoss.1 . . . . . . . 8 (𝜑 → (𝐹𝑌) = 𝑍)
2524adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → (𝐹𝑌) = 𝑍)
2621, 23, 253eqtrd 2778 . . . . . 6 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → ((𝐹𝐺)‘𝑘) = 𝑍)
2726ex 412 . . . . 5 (𝜑 → ((𝑘𝐵 ∧ (𝐺𝑘) = 𝑌) → ((𝐹𝐺)‘𝑘) = 𝑍))
2818, 27biimtrid 242 . . . 4 (𝜑 → ((𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌) → ((𝐹𝐺)‘𝑘) = 𝑍))
2916, 28sylbid 240 . . 3 (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) → ((𝐹𝐺)‘𝑘) = 𝑍))
3029imp 406 . 2 ((𝜑𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌))) → ((𝐹𝐺)‘𝑘) = 𝑍)
315, 30suppss 8217 1 (𝜑 → ((𝐹𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wne 2937  cdif 3959  wss 3962  ran crn 5689  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430   supp csupp 8183
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-supp 8184
This theorem is referenced by:  mplsubglem  22036  mhpinvcl  22173
  Copyright terms: Public domain W3C validator