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Theorem fsuppco2 9162
Description: The composition of a function which maps the zero to zero with a finitely supported function is finitely supported. This is not only a special case of fsuppcor 9163 because it does not require that the "zero" is an element of the range of the finitely supported function. (Contributed by AV, 6-Jun-2019.)
Hypotheses
Ref Expression
fsuppco2.z (𝜑𝑍𝑊)
fsuppco2.f (𝜑𝐹:𝐴𝐵)
fsuppco2.g (𝜑𝐺:𝐵𝐵)
fsuppco2.a (𝜑𝐴𝑈)
fsuppco2.b (𝜑𝐵𝑉)
fsuppco2.n (𝜑𝐹 finSupp 𝑍)
fsuppco2.i (𝜑 → (𝐺𝑍) = 𝑍)
Assertion
Ref Expression
fsuppco2 (𝜑 → (𝐺𝐹) finSupp 𝑍)

Proof of Theorem fsuppco2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fsuppco2.g . . . 4 (𝜑𝐺:𝐵𝐵)
21ffund 6604 . . 3 (𝜑 → Fun 𝐺)
3 fsuppco2.f . . . 4 (𝜑𝐹:𝐴𝐵)
43ffund 6604 . . 3 (𝜑 → Fun 𝐹)
5 funco 6474 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
62, 4, 5syl2anc 584 . 2 (𝜑 → Fun (𝐺𝐹))
7 fsuppco2.n . . . 4 (𝜑𝐹 finSupp 𝑍)
87fsuppimpd 9135 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9 fco 6624 . . . . 5 ((𝐺:𝐵𝐵𝐹:𝐴𝐵) → (𝐺𝐹):𝐴𝐵)
101, 3, 9syl2anc 584 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐵)
11 eldifi 4061 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
12 fvco3 6867 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
133, 11, 12syl2an 596 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
14 ssidd 3944 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
15 fsuppco2.a . . . . . . 7 (𝜑𝐴𝑈)
16 fsuppco2.z . . . . . . 7 (𝜑𝑍𝑊)
173, 14, 15, 16suppssr 8012 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
1817fveq2d 6778 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
19 fsuppco2.i . . . . . 6 (𝜑 → (𝐺𝑍) = 𝑍)
2019adantr 481 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 𝑍)
2113, 18, 203eqtrd 2782 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 𝑍)
2210, 21suppss 8010 . . 3 (𝜑 → ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
238, 22ssfid 9042 . 2 (𝜑 → ((𝐺𝐹) supp 𝑍) ∈ Fin)
24 fsuppco2.b . . . . 5 (𝜑𝐵𝑉)
251, 24fexd 7103 . . . 4 (𝜑𝐺 ∈ V)
263, 15fexd 7103 . . . 4 (𝜑𝐹 ∈ V)
27 coexg 7776 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
2825, 26, 27syl2anc 584 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
29 isfsupp 9132 . . 3 (((𝐺𝐹) ∈ V ∧ 𝑍𝑊) → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
3028, 16, 29syl2anc 584 . 2 (𝜑 → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
316, 23, 30mpbir2and 710 1 (𝜑 → (𝐺𝐹) finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  cdif 3884   class class class wbr 5074  ccom 5593  Fun wfun 6427  wf 6429  cfv 6433  (class class class)co 7275   supp csupp 7977  Fincfn 8733   finSupp cfsupp 9128
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-supp 7978  df-1o 8297  df-en 8734  df-fin 8737  df-fsupp 9129
This theorem is referenced by:  gsumzinv  19546  gsumsub  19549
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