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Theorem fsuppco2 9400
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 9401 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 6721 . . 3 (𝜑 → Fun 𝐺)
3 fsuppco2.f . . . 4 (𝜑𝐹:𝐴𝐵)
43ffund 6721 . . 3 (𝜑 → Fun 𝐹)
5 funco 6588 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
62, 4, 5syl2anc 584 . 2 (𝜑 → Fun (𝐺𝐹))
7 fsuppco2.n . . . 4 (𝜑𝐹 finSupp 𝑍)
87fsuppimpd 9371 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9 fco 6741 . . . . 5 ((𝐺:𝐵𝐵𝐹:𝐴𝐵) → (𝐺𝐹):𝐴𝐵)
101, 3, 9syl2anc 584 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐵)
11 eldifi 4126 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
12 fvco3 6990 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
133, 11, 12syl2an 596 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
14 ssidd 4005 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
15 fsuppco2.a . . . . . . 7 (𝜑𝐴𝑈)
16 fsuppco2.z . . . . . . 7 (𝜑𝑍𝑊)
173, 14, 15, 16suppssr 8183 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
1817fveq2d 6895 . . . . 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 2776 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 𝑍)
2210, 21suppss 8181 . . 3 (𝜑 → ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
238, 22ssfid 9269 . 2 (𝜑 → ((𝐺𝐹) supp 𝑍) ∈ Fin)
24 fsuppco2.b . . . . 5 (𝜑𝐵𝑉)
251, 24fexd 7231 . . . 4 (𝜑𝐺 ∈ V)
263, 15fexd 7231 . . . 4 (𝜑𝐹 ∈ V)
27 coexg 7922 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
2825, 26, 27syl2anc 584 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
29 isfsupp 9367 . . 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 711 1 (𝜑 → (𝐺𝐹) finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  Vcvv 3474  cdif 3945   class class class wbr 5148  ccom 5680  Fun wfun 6537  wf 6539  cfv 6543  (class class class)co 7411   supp csupp 8148  Fincfn 8941   finSupp cfsupp 9363
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-supp 8149  df-1o 8468  df-en 8942  df-fin 8945  df-fsupp 9364
This theorem is referenced by:  gsumzinv  19854  gsumsub  19857
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