Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fsuppinisegfi Structured version   Visualization version   GIF version

Theorem fsuppinisegfi 30450
 Description: The initial segment (◡𝐹 “ {𝑌}) of a nonzero 𝑌 is finite if 𝐹 has finite support. (Contributed by Thierry Arnoux, 21-Jun-2024.)
Hypotheses
Ref Expression
fsuppinisegfi.1 (𝜑𝐹𝑉)
fsuppinisegfi.2 (𝜑0𝑊)
fsuppinisegfi.3 (𝜑𝑌 ∈ (V ∖ { 0 }))
fsuppinisegfi.4 (𝜑𝐹 finSupp 0 )
Assertion
Ref Expression
fsuppinisegfi (𝜑 → (𝐹 “ {𝑌}) ∈ Fin)

Proof of Theorem fsuppinisegfi
StepHypRef Expression
1 fsuppinisegfi.4 . . 3 (𝜑𝐹 finSupp 0 )
21fsuppimpd 8828 . 2 (𝜑 → (𝐹 supp 0 ) ∈ Fin)
3 fsuppinisegfi.3 . . . . 5 (𝜑𝑌 ∈ (V ∖ { 0 }))
43snssd 4705 . . . 4 (𝜑 → {𝑌} ⊆ (V ∖ { 0 }))
5 imass2 5936 . . . 4 ({𝑌} ⊆ (V ∖ { 0 }) → (𝐹 “ {𝑌}) ⊆ (𝐹 “ (V ∖ { 0 })))
64, 5syl 17 . . 3 (𝜑 → (𝐹 “ {𝑌}) ⊆ (𝐹 “ (V ∖ { 0 })))
7 fsuppinisegfi.1 . . . 4 (𝜑𝐹𝑉)
8 fsuppinisegfi.2 . . . 4 (𝜑0𝑊)
9 suppimacnvss 7827 . . . 4 ((𝐹𝑉0𝑊) → (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 supp 0 ))
107, 8, 9syl2anc 587 . . 3 (𝜑 → (𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 supp 0 ))
116, 10sstrd 3928 . 2 (𝜑 → (𝐹 “ {𝑌}) ⊆ (𝐹 supp 0 ))
122, 11ssfid 8729 1 (𝜑 → (𝐹 “ {𝑌}) ∈ Fin)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112  Vcvv 3444   ∖ cdif 3881   ⊆ wss 3884  {csn 4528   class class class wbr 5033  ◡ccnv 5522   “ cima 5526  (class class class)co 7139   supp csupp 7817  Fincfn 8496   finSupp cfsupp 8821 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-supp 7818  df-er 8276  df-en 8497  df-fin 8500  df-fsupp 8822 This theorem is referenced by:  elrspunidl  31017
 Copyright terms: Public domain W3C validator