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Theorem fsuppcor 9444
Description: The composition of a function which maps the zero of the range of a finitely supported function to the zero of its range with this finitely supported function is finitely supported. (Contributed by AV, 6-Jun-2019.)
Hypotheses
Ref Expression
fsuppcor.0 (𝜑0𝑊)
fsuppcor.z (𝜑𝑍𝐵)
fsuppcor.f (𝜑𝐹:𝐴𝐶)
fsuppcor.g (𝜑𝐺:𝐵𝐷)
fsuppcor.s (𝜑𝐶𝐵)
fsuppcor.a (𝜑𝐴𝑈)
fsuppcor.b (𝜑𝐵𝑉)
fsuppcor.n (𝜑𝐹 finSupp 𝑍)
fsuppcor.i (𝜑 → (𝐺𝑍) = 0 )
Assertion
Ref Expression
fsuppcor (𝜑 → (𝐺𝐹) finSupp 0 )

Proof of Theorem fsuppcor
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fsuppcor.g . . . 4 (𝜑𝐺:𝐵𝐷)
21ffund 6740 . . 3 (𝜑 → Fun 𝐺)
3 fsuppcor.f . . . 4 (𝜑𝐹:𝐴𝐶)
43ffund 6740 . . 3 (𝜑 → Fun 𝐹)
5 funco 6606 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
62, 4, 5syl2anc 584 . 2 (𝜑 → Fun (𝐺𝐹))
7 fsuppcor.n . . . 4 (𝜑𝐹 finSupp 𝑍)
87fsuppimpd 9409 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9 fsuppcor.s . . . . . 6 (𝜑𝐶𝐵)
101, 9fssresd 6775 . . . . 5 (𝜑 → (𝐺𝐶):𝐶𝐷)
11 fco2 6762 . . . . 5 (((𝐺𝐶):𝐶𝐷𝐹:𝐴𝐶) → (𝐺𝐹):𝐴𝐷)
1210, 3, 11syl2anc 584 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐷)
13 eldifi 4131 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
14 fvco3 7008 . . . . . 6 ((𝐹:𝐴𝐶𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
153, 13, 14syl2an 596 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
16 ssidd 4007 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
17 fsuppcor.a . . . . . . 7 (𝜑𝐴𝑈)
18 fsuppcor.z . . . . . . 7 (𝜑𝑍𝐵)
193, 16, 17, 18suppssr 8220 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
2019fveq2d 6910 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
21 fsuppcor.i . . . . . 6 (𝜑 → (𝐺𝑍) = 0 )
2221adantr 480 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 0 )
2315, 20, 223eqtrd 2781 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 0 )
2412, 23suppss 8219 . . 3 (𝜑 → ((𝐺𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
258, 24ssfid 9301 . 2 (𝜑 → ((𝐺𝐹) supp 0 ) ∈ Fin)
26 fsuppcor.b . . . . 5 (𝜑𝐵𝑉)
271, 26fexd 7247 . . . 4 (𝜑𝐺 ∈ V)
283, 17fexd 7247 . . . 4 (𝜑𝐹 ∈ V)
29 coexg 7951 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3027, 28, 29syl2anc 584 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
31 fsuppcor.0 . . 3 (𝜑0𝑊)
32 isfsupp 9405 . . 3 (((𝐺𝐹) ∈ V ∧ 0𝑊) → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
3330, 31, 32syl2anc 584 . 2 (𝜑 → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
346, 25, 33mpbir2and 713 1 (𝜑 → (𝐺𝐹) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  wss 3951   class class class wbr 5143  cres 5687  ccom 5689  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431   supp csupp 8185  Fincfn 8985   finSupp cfsupp 9401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-supp 8186  df-1o 8506  df-en 8986  df-fin 8989  df-fsupp 9402
This theorem is referenced by:  mapfienlem1  9445  mapfienlem2  9446  mhmcompl  22384  cpmadumatpolylem2  22888  selvvvval  42595
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