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Theorem suppcoss 8130
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 8125, 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 6678 . . . 4 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
31, 2sylib 217 . . 3 (𝜑𝐹:𝐴⟶ran 𝐹)
4 suppcoss.g . . 3 (𝜑𝐺:𝐵𝐴)
53, 4fcod 6691 . 2 (𝜑 → (𝐹𝐺):𝐵⟶ran 𝐹)
6 eldif 3918 . . . . 5 (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)))
74ffnd 6666 . . . . . . . . 9 (𝜑𝐺 Fn 𝐵)
8 suppcoss.b . . . . . . . . 9 (𝜑𝐵𝑊)
9 suppcoss.y . . . . . . . . 9 (𝜑𝑌𝑉)
10 elsuppfn 8094 . . . . . . . . 9 ((𝐺 Fn 𝐵𝐵𝑊𝑌𝑉) → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
117, 8, 9, 10syl3anc 1371 . . . . . . . 8 (𝜑 → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
1211notbid 317 . . . . . . 7 (𝜑 → (¬ 𝑘 ∈ (𝐺 supp 𝑌) ↔ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
1312anbi2d 629 . . . . . 6 (𝜑 → ((𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌))))
14 annotanannot 833 . . . . . 6 ((𝑘𝐵 ∧ ¬ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌))
1513, 14bitrdi 286 . . . . 5 (𝜑 → ((𝑘𝐵 ∧ ¬ 𝑘 ∈ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌)))
166, 15bitrid 282 . . . 4 (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) ↔ (𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌)))
17 nne 2945 . . . . . 6 (¬ (𝐺𝑘) ≠ 𝑌 ↔ (𝐺𝑘) = 𝑌)
1817anbi2i 623 . . . . 5 ((𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌))
194adantr 481 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → 𝐺:𝐵𝐴)
20 simprl 769 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → 𝑘𝐵)
2119, 20fvco3d 6938 . . . . . . 7 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → ((𝐹𝐺)‘𝑘) = (𝐹‘(𝐺𝑘)))
22 simprr 771 . . . . . . . 8 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → (𝐺𝑘) = 𝑌)
2322fveq2d 6843 . . . . . . 7 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → (𝐹‘(𝐺𝑘)) = (𝐹𝑌))
24 suppcoss.1 . . . . . . . 8 (𝜑 → (𝐹𝑌) = 𝑍)
2524adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → (𝐹𝑌) = 𝑍)
2621, 23, 253eqtrd 2781 . . . . . 6 ((𝜑 ∧ (𝑘𝐵 ∧ (𝐺𝑘) = 𝑌)) → ((𝐹𝐺)‘𝑘) = 𝑍)
2726ex 413 . . . . 5 (𝜑 → ((𝑘𝐵 ∧ (𝐺𝑘) = 𝑌) → ((𝐹𝐺)‘𝑘) = 𝑍))
2818, 27biimtrid 241 . . . 4 (𝜑 → ((𝑘𝐵 ∧ ¬ (𝐺𝑘) ≠ 𝑌) → ((𝐹𝐺)‘𝑘) = 𝑍))
2916, 28sylbid 239 . . 3 (𝜑 → (𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌)) → ((𝐹𝐺)‘𝑘) = 𝑍))
3029imp 407 . 2 ((𝜑𝑘 ∈ (𝐵 ∖ (𝐺 supp 𝑌))) → ((𝐹𝐺)‘𝑘) = 𝑍)
315, 30suppss 8117 1 (𝜑 → ((𝐹𝐺) supp 𝑍) ⊆ (𝐺 supp 𝑌))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wne 2941  cdif 3905  wss 3908  ran crn 5632  ccom 5635   Fn wfn 6488  wf 6489  cfv 6493  (class class class)co 7351   supp csupp 8084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-supp 8085
This theorem is referenced by:  mplsubglem  21357  mhpinvcl  21494
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