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Theorem suppcoss 8232
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 8227, 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 3961 . . . . 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 8195 . . . . . . . . 9 ((𝐺 Fn 𝐵𝐵𝑊𝑌𝑉) → (𝑘 ∈ (𝐺 supp 𝑌) ↔ (𝑘𝐵 ∧ (𝐺𝑘) ≠ 𝑌)))
117, 8, 9, 10syl3anc 1373 . . . . . . . 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 2944 . . . . . 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 7009 . . . . . . 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 2781 . . . . . 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 8219 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 1540  wcel 2108  wne 2940  cdif 3948  wss 3951  ran crn 5686  ccom 5689   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431   supp csupp 8185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-supp 8186
This theorem is referenced by:  mplsubglem  22019  mhpinvcl  22156
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