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Theorem fsuppcor 9347
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 6677 . . 3 (𝜑 → Fun 𝐺)
3 fsuppcor.f . . . 4 (𝜑𝐹:𝐴𝐶)
43ffund 6677 . . 3 (𝜑 → Fun 𝐹)
5 funco 6546 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
62, 4, 5syl2anc 585 . 2 (𝜑 → Fun (𝐺𝐹))
7 fsuppcor.n . . . 4 (𝜑𝐹 finSupp 𝑍)
87fsuppimpd 9319 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9 fsuppcor.s . . . . . 6 (𝜑𝐶𝐵)
101, 9fssresd 6714 . . . . 5 (𝜑 → (𝐺𝐶):𝐶𝐷)
11 fco2 6700 . . . . 5 (((𝐺𝐶):𝐶𝐷𝐹:𝐴𝐶) → (𝐺𝐹):𝐴𝐷)
1210, 3, 11syl2anc 585 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐷)
13 eldifi 4091 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
14 fvco3 6945 . . . . . 6 ((𝐹:𝐴𝐶𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
153, 13, 14syl2an 597 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
16 ssidd 3972 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
17 fsuppcor.a . . . . . . 7 (𝜑𝐴𝑈)
18 fsuppcor.z . . . . . . 7 (𝜑𝑍𝐵)
193, 16, 17, 18suppssr 8132 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
2019fveq2d 6851 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
21 fsuppcor.i . . . . . 6 (𝜑 → (𝐺𝑍) = 0 )
2221adantr 482 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 0 )
2315, 20, 223eqtrd 2781 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 0 )
2412, 23suppss 8130 . . 3 (𝜑 → ((𝐺𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
258, 24ssfid 9218 . 2 (𝜑 → ((𝐺𝐹) supp 0 ) ∈ Fin)
26 fsuppcor.b . . . . 5 (𝜑𝐵𝑉)
271, 26fexd 7182 . . . 4 (𝜑𝐺 ∈ V)
283, 17fexd 7182 . . . 4 (𝜑𝐹 ∈ V)
29 coexg 7871 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3027, 28, 29syl2anc 585 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
31 fsuppcor.0 . . 3 (𝜑0𝑊)
32 isfsupp 9316 . . 3 (((𝐺𝐹) ∈ V ∧ 0𝑊) → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
3330, 31, 32syl2anc 585 . 2 (𝜑 → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
346, 25, 33mpbir2and 712 1 (𝜑 → (𝐺𝐹) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3448  cdif 3912  wss 3915   class class class wbr 5110  cres 5640  ccom 5642  Fun wfun 6495  wf 6497  cfv 6501  (class class class)co 7362   supp csupp 8097  Fincfn 8890   finSupp cfsupp 9312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-supp 8098  df-1o 8417  df-en 8891  df-fin 8894  df-fsupp 9313
This theorem is referenced by:  mapfienlem1  9348  mapfienlem2  9349  cpmadumatpolylem2  22247  mhmcompl  40765
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