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Theorem offinsupp1 30586
 Description: Finite support for a function operation. (Contributed by Thierry Arnoux, 8-Jul-2023.)
Hypotheses
Ref Expression
offinsupp1.a (𝜑𝐴𝑉)
offinsupp1.y (𝜑𝑌𝑈)
offinsupp1.z (𝜑𝑍𝑊)
offinsupp1.f (𝜑𝐹:𝐴𝑆)
offinsupp1.g (𝜑𝐺:𝐴𝑇)
offinsupp1.1 (𝜑𝐹 finSupp 𝑌)
offinsupp1.2 ((𝜑𝑥𝑇) → (𝑌𝑅𝑥) = 𝑍)
Assertion
Ref Expression
offinsupp1 (𝜑 → (𝐹f 𝑅𝐺) finSupp 𝑍)
Distinct variable groups:   𝑥,𝐺   𝑥,𝑅   𝑥,𝑇   𝑥,𝑌   𝑥,𝑍   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝑆(𝑥)   𝑈(𝑥)   𝐹(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem offinsupp1
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 offinsupp1.1 . . . 4 (𝜑𝐹 finSupp 𝑌)
21fsuppimpd 8873 . . 3 (𝜑 → (𝐹 supp 𝑌) ∈ Fin)
3 ssidd 3915 . . . 4 (𝜑 → (𝐹 supp 𝑌) ⊆ (𝐹 supp 𝑌))
4 offinsupp1.2 . . . 4 ((𝜑𝑥𝑇) → (𝑌𝑅𝑥) = 𝑍)
5 offinsupp1.f . . . 4 (𝜑𝐹:𝐴𝑆)
6 offinsupp1.g . . . 4 (𝜑𝐺:𝐴𝑇)
7 offinsupp1.a . . . 4 (𝜑𝐴𝑉)
8 offinsupp1.y . . . 4 (𝜑𝑌𝑈)
93, 4, 5, 6, 7, 8suppssof1 7873 . . 3 (𝜑 → ((𝐹f 𝑅𝐺) supp 𝑍) ⊆ (𝐹 supp 𝑌))
102, 9ssfid 8778 . 2 (𝜑 → ((𝐹f 𝑅𝐺) supp 𝑍) ∈ Fin)
11 ovexd 7185 . . . . 5 ((𝜑 ∧ (𝑖𝑆𝑗𝑇)) → (𝑖𝑅𝑗) ∈ V)
12 inidm 4123 . . . . 5 (𝐴𝐴) = 𝐴
1311, 5, 6, 7, 7, 12off 7422 . . . 4 (𝜑 → (𝐹f 𝑅𝐺):𝐴⟶V)
1413ffund 6502 . . 3 (𝜑 → Fun (𝐹f 𝑅𝐺))
15 ovexd 7185 . . 3 (𝜑 → (𝐹f 𝑅𝐺) ∈ V)
16 offinsupp1.z . . 3 (𝜑𝑍𝑊)
17 funisfsupp 8871 . . 3 ((Fun (𝐹f 𝑅𝐺) ∧ (𝐹f 𝑅𝐺) ∈ V ∧ 𝑍𝑊) → ((𝐹f 𝑅𝐺) finSupp 𝑍 ↔ ((𝐹f 𝑅𝐺) supp 𝑍) ∈ Fin))
1814, 15, 16, 17syl3anc 1368 . 2 (𝜑 → ((𝐹f 𝑅𝐺) finSupp 𝑍 ↔ ((𝐹f 𝑅𝐺) supp 𝑍) ∈ Fin))
1910, 18mpbird 260 1 (𝜑 → (𝐹f 𝑅𝐺) finSupp 𝑍)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3409   class class class wbr 5032  Fun wfun 6329  ⟶wf 6331  (class class class)co 7150   ∘f cof 7403   supp csupp 7835  Fincfn 8527   finSupp cfsupp 8866 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-of 7405  df-om 7580  df-supp 7836  df-1o 8112  df-en 8528  df-fin 8531  df-fsupp 8867 This theorem is referenced by:  fedgmullem1  31231
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