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Theorem fsuppco2 8941
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 8942 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 6509 . . 3 (𝜑 → Fun 𝐺)
3 fsuppco2.f . . . 4 (𝜑𝐹:𝐴𝐵)
43ffund 6509 . . 3 (𝜑 → Fun 𝐹)
5 funco 6380 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
62, 4, 5syl2anc 587 . 2 (𝜑 → Fun (𝐺𝐹))
7 fsuppco2.n . . . 4 (𝜑𝐹 finSupp 𝑍)
87fsuppimpd 8914 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9 fco 6529 . . . . 5 ((𝐺:𝐵𝐵𝐹:𝐴𝐵) → (𝐺𝐹):𝐴𝐵)
101, 3, 9syl2anc 587 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐵)
11 eldifi 4018 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
12 fvco3 6768 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
133, 11, 12syl2an 599 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
14 ssidd 3901 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
15 fsuppco2.a . . . . . . 7 (𝜑𝐴𝑈)
16 fsuppco2.z . . . . . . 7 (𝜑𝑍𝑊)
173, 14, 15, 16suppssr 7892 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
1817fveq2d 6679 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
19 fsuppco2.i . . . . . 6 (𝜑 → (𝐺𝑍) = 𝑍)
2019adantr 484 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 𝑍)
2113, 18, 203eqtrd 2777 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 𝑍)
2210, 21suppss 7890 . . 3 (𝜑 → ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
238, 22ssfid 8820 . 2 (𝜑 → ((𝐺𝐹) supp 𝑍) ∈ Fin)
24 fsuppco2.b . . . . 5 (𝜑𝐵𝑉)
251, 24fexd 7001 . . . 4 (𝜑𝐺 ∈ V)
263, 15fexd 7001 . . . 4 (𝜑𝐹 ∈ V)
27 coexg 7661 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
2825, 26, 27syl2anc 587 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
29 isfsupp 8911 . . 3 (((𝐺𝐹) ∈ V ∧ 𝑍𝑊) → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
3028, 16, 29syl2anc 587 . 2 (𝜑 → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
316, 23, 30mpbir2and 713 1 (𝜑 → (𝐺𝐹) finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2113  Vcvv 3398  cdif 3841   class class class wbr 5031  ccom 5530  Fun wfun 6334  wf 6336  cfv 6340  (class class class)co 7171   supp csupp 7857  Fincfn 8556   finSupp cfsupp 8907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7480
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3683  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-pss 3863  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-tp 4522  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5484  df-we 5486  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7174  df-oprab 7175  df-mpo 7176  df-om 7601  df-supp 7858  df-1o 8132  df-en 8557  df-fin 8560  df-fsupp 8908
This theorem is referenced by:  gsumzinv  19185  gsumsub  19188
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