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Theorem fsuppco2 8293
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 8294 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 6035 . . . 4 (𝐺:𝐵𝐵 → Fun 𝐺)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐺)
4 fsuppco2.f . . . 4 (𝜑𝐹:𝐴𝐵)
5 ffun 6035 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
64, 5syl 17 . . 3 (𝜑 → Fun 𝐹)
7 funco 5916 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
83, 6, 7syl2anc 692 . 2 (𝜑 → Fun (𝐺𝐹))
9 fsuppco2.n . . . 4 (𝜑𝐹 finSupp 𝑍)
109fsuppimpd 8267 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
11 fco 6045 . . . . 5 ((𝐺:𝐵𝐵𝐹:𝐴𝐵) → (𝐺𝐹):𝐴𝐵)
121, 4, 11syl2anc 692 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐵)
13 eldifi 3724 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
14 fvco3 6262 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
154, 13, 14syl2an 494 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
16 ssid 3616 . . . . . . . 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 7311 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
2120fveq2d 6182 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
22 fsuppco2.i . . . . . 6 (𝜑 → (𝐺𝑍) = 𝑍)
2322adantr 481 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 𝑍)
2415, 21, 233eqtrd 2658 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 𝑍)
2512, 24suppss 7310 . . 3 (𝜑 → ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
26 ssfi 8165 . . 3 (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐺𝐹) supp 𝑍) ∈ Fin)
2710, 25, 26syl2anc 692 . 2 (𝜑 → ((𝐺𝐹) supp 𝑍) ∈ Fin)
28 fsuppco2.b . . . . 5 (𝜑𝐵𝑉)
29 fex 6475 . . . . 5 ((𝐺:𝐵𝐵𝐵𝑉) → 𝐺 ∈ V)
301, 28, 29syl2anc 692 . . . 4 (𝜑𝐺 ∈ V)
31 fex 6475 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑈) → 𝐹 ∈ V)
324, 18, 31syl2anc 692 . . . 4 (𝜑𝐹 ∈ V)
33 coexg 7102 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3430, 32, 33syl2anc 692 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
35 isfsupp 8264 . . 3 (((𝐺𝐹) ∈ V ∧ 𝑍𝑊) → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
3634, 19, 35syl2anc 692 . 2 (𝜑 → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
378, 27, 36mpbir2and 956 1 (𝜑 → (𝐺𝐹) finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cdif 3564  wss 3567   class class class wbr 4644  ccom 5108  Fun wfun 5870  wf 5872  cfv 5876  (class class class)co 6635   supp csupp 7280  Fincfn 7940   finSupp cfsupp 8260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-supp 7281  df-er 7727  df-en 7941  df-fin 7944  df-fsupp 8261
This theorem is referenced by:  gsumzinv  18326  gsumsub  18329
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