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Theorem fsuppcor 9401
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 6715 . . 3 (𝜑 → Fun 𝐺)
3 fsuppcor.f . . . 4 (𝜑𝐹:𝐴𝐶)
43ffund 6715 . . 3 (𝜑 → Fun 𝐹)
5 funco 6582 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
62, 4, 5syl2anc 583 . 2 (𝜑 → Fun (𝐺𝐹))
7 fsuppcor.n . . . 4 (𝜑𝐹 finSupp 𝑍)
87fsuppimpd 9371 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9 fsuppcor.s . . . . . 6 (𝜑𝐶𝐵)
101, 9fssresd 6752 . . . . 5 (𝜑 → (𝐺𝐶):𝐶𝐷)
11 fco2 6738 . . . . 5 (((𝐺𝐶):𝐶𝐷𝐹:𝐴𝐶) → (𝐺𝐹):𝐴𝐷)
1210, 3, 11syl2anc 583 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐷)
13 eldifi 4121 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
14 fvco3 6984 . . . . . 6 ((𝐹:𝐴𝐶𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
153, 13, 14syl2an 595 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
16 ssidd 4000 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
17 fsuppcor.a . . . . . . 7 (𝜑𝐴𝑈)
18 fsuppcor.z . . . . . . 7 (𝜑𝑍𝐵)
193, 16, 17, 18suppssr 8181 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
2019fveq2d 6889 . . . . 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 2770 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 0 )
2412, 23suppss 8179 . . 3 (𝜑 → ((𝐺𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
258, 24ssfid 9269 . 2 (𝜑 → ((𝐺𝐹) supp 0 ) ∈ Fin)
26 fsuppcor.b . . . . 5 (𝜑𝐵𝑉)
271, 26fexd 7224 . . . 4 (𝜑𝐺 ∈ V)
283, 17fexd 7224 . . . 4 (𝜑𝐹 ∈ V)
29 coexg 7919 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3027, 28, 29syl2anc 583 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
31 fsuppcor.0 . . 3 (𝜑0𝑊)
32 isfsupp 9367 . . 3 (((𝐺𝐹) ∈ V ∧ 0𝑊) → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
3330, 31, 32syl2anc 583 . 2 (𝜑 → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
346, 25, 33mpbir2and 710 1 (𝜑 → (𝐺𝐹) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  cdif 3940  wss 3943   class class class wbr 5141  cres 5671  ccom 5673  Fun wfun 6531  wf 6533  cfv 6537  (class class class)co 7405   supp csupp 8146  Fincfn 8941   finSupp cfsupp 9363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-supp 8147  df-1o 8467  df-en 8942  df-fin 8945  df-fsupp 9364
This theorem is referenced by:  mapfienlem1  9402  mapfienlem2  9403  cpmadumatpolylem2  22739  mhmcompl  41677  selvvvval  41714
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