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Theorem fsuppco2 8556
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 8557 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 6269 . . 3 (𝜑 → Fun 𝐺)
3 fsuppco2.f . . . 4 (𝜑𝐹:𝐴𝐵)
43ffund 6269 . . 3 (𝜑 → Fun 𝐹)
5 funco 6150 . . 3 ((Fun 𝐺 ∧ Fun 𝐹) → Fun (𝐺𝐹))
62, 4, 5syl2anc 575 . 2 (𝜑 → Fun (𝐺𝐹))
7 fsuppco2.n . . . 4 (𝜑𝐹 finSupp 𝑍)
87fsuppimpd 8530 . . 3 (𝜑 → (𝐹 supp 𝑍) ∈ Fin)
9 fco 6282 . . . . 5 ((𝐺:𝐵𝐵𝐹:𝐴𝐵) → (𝐺𝐹):𝐴𝐵)
101, 3, 9syl2anc 575 . . . 4 (𝜑 → (𝐺𝐹):𝐴𝐵)
11 eldifi 3942 . . . . . 6 (𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍)) → 𝑥𝐴)
12 fvco3 6505 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
133, 11, 12syl2an 585 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
14 ssidd 3832 . . . . . . 7 (𝜑 → (𝐹 supp 𝑍) ⊆ (𝐹 supp 𝑍))
15 fsuppco2.a . . . . . . 7 (𝜑𝐴𝑈)
16 fsuppco2.z . . . . . . 7 (𝜑𝑍𝑊)
173, 14, 15, 16suppssr 7570 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐹𝑥) = 𝑍)
1817fveq2d 6421 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺‘(𝐹𝑥)) = (𝐺𝑍))
19 fsuppco2.i . . . . . 6 (𝜑 → (𝐺𝑍) = 𝑍)
2019adantr 468 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → (𝐺𝑍) = 𝑍)
2113, 18, 203eqtrd 2855 . . . 4 ((𝜑𝑥 ∈ (𝐴 ∖ (𝐹 supp 𝑍))) → ((𝐺𝐹)‘𝑥) = 𝑍)
2210, 21suppss 7569 . . 3 (𝜑 → ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍))
23 ssfi 8428 . . 3 (((𝐹 supp 𝑍) ∈ Fin ∧ ((𝐺𝐹) supp 𝑍) ⊆ (𝐹 supp 𝑍)) → ((𝐺𝐹) supp 𝑍) ∈ Fin)
248, 22, 23syl2anc 575 . 2 (𝜑 → ((𝐺𝐹) supp 𝑍) ∈ Fin)
25 fsuppco2.b . . . . 5 (𝜑𝐵𝑉)
26 fex 6723 . . . . 5 ((𝐺:𝐵𝐵𝐵𝑉) → 𝐺 ∈ V)
271, 25, 26syl2anc 575 . . . 4 (𝜑𝐺 ∈ V)
28 fex 6723 . . . . 5 ((𝐹:𝐴𝐵𝐴𝑈) → 𝐹 ∈ V)
293, 15, 28syl2anc 575 . . . 4 (𝜑𝐹 ∈ V)
30 coexg 7356 . . . 4 ((𝐺 ∈ V ∧ 𝐹 ∈ V) → (𝐺𝐹) ∈ V)
3127, 29, 30syl2anc 575 . . 3 (𝜑 → (𝐺𝐹) ∈ V)
32 isfsupp 8527 . . 3 (((𝐺𝐹) ∈ V ∧ 𝑍𝑊) → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
3331, 16, 32syl2anc 575 . 2 (𝜑 → ((𝐺𝐹) finSupp 𝑍 ↔ (Fun (𝐺𝐹) ∧ ((𝐺𝐹) supp 𝑍) ∈ Fin)))
346, 24, 33mpbir2and 695 1 (𝜑 → (𝐺𝐹) finSupp 𝑍)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2157  Vcvv 3402  cdif 3777  wss 3780   class class class wbr 4855  ccom 5328  Fun wfun 6104  wf 6106  cfv 6110  (class class class)co 6883   supp csupp 7538  Fincfn 8201   finSupp cfsupp 8523
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 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7188
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 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-iun 4725  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-ord 5952  df-on 5953  df-lim 5954  df-suc 5955  df-iota 6073  df-fun 6112  df-fn 6113  df-f 6114  df-f1 6115  df-fo 6116  df-f1o 6117  df-fv 6118  df-ov 6886  df-oprab 6887  df-mpt2 6888  df-om 7305  df-supp 7539  df-er 7988  df-en 8202  df-fin 8205  df-fsupp 8524
This theorem is referenced by:  gsumzinv  18565  gsumsub  18568
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