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Theorem fsuppco2 8191
 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 8192 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)
2 ffun 5961 . . . 4 (𝐺:𝐵𝐵 → Fun 𝐺)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐺)
4 fsuppco2.f . . . 4 (𝜑𝐹:𝐴𝐵)
5 ffun 5961 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
64, 5syl 17 . . 3 (𝜑 → Fun 𝐹)
7 funco 5842 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
83, 6, 7syl2anc 691 . 2 (𝜑 → Fun (𝐺𝐹))
9 fsuppco2.n . . . 4 (𝜑𝐹 finSupp 𝑍)
109fsuppimpd 8165 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
11 fco 5971 . . . . 5 ((𝐺:𝐵𝐵𝐹:𝐴𝐵) → (𝐺𝐹):𝐴𝐵)
121, 4, 11syl2anc 691 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐵)
13 eldifi 3694 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
14 fvco3 6185 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
154, 13, 14syl2an 493 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
16 ssid 3587 . . . . . . . 8 (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍)
1716a1i 11 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
18 fsuppco2.a . . . . . . 7 (𝜑𝐴𝑈)
19 fsuppco2.z . . . . . . 7 (𝜑𝑍𝑊)
204, 17, 18, 19suppssr 7213 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
2120fveq2d 6107 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
22 fsuppco2.i . . . . . 6 (𝜑 → (𝐺𝑍) = 𝑍)
2322adantr 480 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 𝑍)
2415, 21, 233eqtrd 2648 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 𝑍)
2512, 24suppss 7212 . . 3 (𝜑 → ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
26 ssfi 8065 . . 3 (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐺𝐹) supp 𝑍) ∈ Fin)
2710, 25, 26syl2anc 691 . 2 (𝜑 → ((𝐺𝐹) supp 𝑍) ∈ Fin)
28 fsuppco2.b . . . . 5 (𝜑𝐵𝑉)
29 fex 6394 . . . . 5 ((𝐺:𝐵𝐵𝐵𝑉) → 𝐺 ∈ V)
301, 28, 29syl2anc 691 . . . 4 (𝜑𝐺 ∈ V)
31 fex 6394 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑈) → 𝐹 ∈ V)
324, 18, 31syl2anc 691 . . . 4 (𝜑𝐹 ∈ V)
33 coexg 7010 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3430, 32, 33syl2anc 691 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
35 isfsupp 8162 . . 3 (((𝐺𝐹) ∈ V ∧ 𝑍𝑊) → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
3634, 19, 35syl2anc 691 . 2 (𝜑 → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
378, 27, 36mpbir2and 959 1 (𝜑 → (𝐺𝐹) finSupp 𝑍)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540   class class class wbr 4583   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   supp csupp 7182  Fincfn 7841   finSupp cfsupp 8158 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-supp 7183  df-er 7629  df-en 7842  df-fin 7845  df-fsupp 8159 This theorem is referenced by:  gsumzinv  18168  gsumsub  18171
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