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Theorem fsuppcor 8542
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 6254 . . 3 (𝜑 → Fun 𝐺)
3 fsuppcor.f . . . 4 (𝜑𝐹:𝐴𝐶)
43ffund 6254 . . 3 (𝜑 → Fun 𝐹)
5 funco 6135 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
62, 4, 5syl2anc 575 . 2 (𝜑 → Fun (𝐺𝐹))
7 fsuppcor.n . . . 4 (𝜑𝐹 finSupp 𝑍)
87fsuppimpd 8515 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9 fsuppcor.s . . . . . 6 (𝜑𝐶𝐵)
101, 9fssresd 6280 . . . . 5 (𝜑 → (𝐺𝐶):𝐶𝐷)
11 fco2 6268 . . . . 5 (((𝐺𝐶):𝐶𝐷𝐹:𝐴𝐶) → (𝐺𝐹):𝐴𝐷)
1210, 3, 11syl2anc 575 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐷)
13 eldifi 3925 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
14 fvco3 6490 . . . . . 6 ((𝐹:𝐴𝐶𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
153, 13, 14syl2an 585 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
16 ssidd 3815 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
17 fsuppcor.a . . . . . . 7 (𝜑𝐴𝑈)
18 fsuppcor.z . . . . . . 7 (𝜑𝑍𝐵)
193, 16, 17, 18suppssr 7555 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
2019fveq2d 6406 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
21 fsuppcor.i . . . . . 6 (𝜑 → (𝐺𝑍) = 0 )
2221adantr 468 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 0 )
2315, 20, 223eqtrd 2840 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 0 )
2412, 23suppss 7554 . . 3 (𝜑 → ((𝐺𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍))
25 ssfi 8413 . . 3 (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺𝐹) supp 0 ) ⊆ (𝐹 supp 𝑍)) → ((𝐺𝐹) supp 0 ) ∈ Fin)
268, 24, 25syl2anc 575 . 2 (𝜑 → ((𝐺𝐹) supp 0 ) ∈ Fin)
27 fsuppcor.b . . . . 5 (𝜑𝐵𝑉)
28 fex 6708 . . . . 5 ((𝐺:𝐵𝐷𝐵𝑉) → 𝐺 ∈ V)
291, 27, 28syl2anc 575 . . . 4 (𝜑𝐺 ∈ V)
30 fex 6708 . . . . 5 ((𝐹:𝐴𝐶𝐴𝑈) → 𝐹 ∈ V)
313, 17, 30syl2anc 575 . . . 4 (𝜑𝐹 ∈ V)
32 coexg 7341 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3329, 31, 32syl2anc 575 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
34 fsuppcor.0 . . 3 (𝜑0𝑊)
35 isfsupp 8512 . . 3 (((𝐺𝐹) ∈ V ∧ 0𝑊) → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
3633, 34, 35syl2anc 575 . 2 (𝜑 → ((𝐺𝐹) finSupp 0 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 0 ) ∈ Fin)))
376, 26, 36mpbir2and 695 1 (𝜑 → (𝐺𝐹) finSupp 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2155  Vcvv 3387  cdif 3760  wss 3763   class class class wbr 4837  cres 5307  ccom 5309  Fun wfun 6089  wf 6091  cfv 6095  (class class class)co 6868   supp csupp 7523  Fincfn 8186   finSupp cfsupp 8508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-rep 4957  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090  ax-un 7173
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-pss 3779  df-nul 4111  df-if 4274  df-pw 4347  df-sn 4365  df-pr 4367  df-tp 4369  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-tr 4940  df-id 5213  df-eprel 5218  df-po 5226  df-so 5227  df-fr 5264  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-ord 5933  df-on 5934  df-lim 5935  df-suc 5936  df-iota 6058  df-fun 6097  df-fn 6098  df-f 6099  df-f1 6100  df-fo 6101  df-f1o 6102  df-fv 6103  df-ov 6871  df-oprab 6872  df-mpt2 6873  df-om 7290  df-supp 7524  df-er 7973  df-en 8187  df-fin 8190  df-fsupp 8509
This theorem is referenced by:  mapfienlem1  8543  mapfienlem2  8544  cpmadumatpolylem2  20894
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