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Theorem fissfi 7229
Description: A finite subset of a finite set is a decidable subset. (Contributed by Jim Kingdon, 18-May-2026.)
Assertion
Ref Expression
fissfi |- ( ( S C_ A /\ A e. Fin /\ S e. Fin ) -> A. x e. A DECID x e. S )
Distinct variable groups:   x, A   x, S
Proof of Theorem fissfi
Dummy variables u v w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2298 . . . 4 |- ( w = (/) -> ( x e. w <-> x e. (/) ) )
21dcbid 846 . . 3 |- ( w = (/) -> (DECID x e. w <-> DECID x e. (/) ) )
3 eleq2 2298 . . . 4 |- ( w = u -> ( x e. w <-> x e. u ) )
43dcbid 846 . . 3 |- ( w = u -> (DECID x e. w <-> DECID x e. u ) )
5 eleq2 2298 . . . 4 |- ( w = ( u u. { v } ) -> ( x e. w <-> x e. ( u u. { v } ) ) )
65dcbid 846 . . 3 |- ( w = ( u u. { v } ) -> (DECID x e. w <-> DECID x e. ( u u. { v } ) ) )
7 eleq2 2298 . . . 4 |- ( w = S -> ( x e. w <-> x e. S ) )
87dcbid 846 . . 3 |- ( w = S -> (DECID x e. w <-> DECID x e. S ) )
9 noel 3516 . . . . . 6 |- -. x e. (/)
109olci 740 . . . . 5 |- ( x e. (/) \/ -. x e. (/) )
11 df-dc 843 . . . . 5 |- (DECID x e. (/) <-> ( x e. (/) \/ -. x e. (/) ) )
1210, 11mpbir 146 . . . 4 |- DECID x e. (/)
1312a1i 9 . . 3 |- ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) -> DECID x e. (/) )
14 simpr 110 . . . . 5 |- ( ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) /\ DECID x e. u ) -> DECID x e. u )
15 simp2 1025 . . . . . . . 8 |- ( ( S C_ A /\ A e. Fin /\ S e. Fin ) -> A e. Fin )
1615ad4antr 494 . . . . . . 7 |- ( ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) /\ DECID x e. u ) -> A e. Fin )
17 simp-4r 544 . . . . . . 7 |- ( ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) /\ DECID x e. u ) -> x e. A )
18 simp1 1024 . . . . . . . . 9 |- ( ( S C_ A /\ A e. Fin /\ S e. Fin ) -> S C_ A )
1918ad4antr 494 . . . . . . . 8 |- ( ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) /\ DECID x e. u ) -> S C_ A )
20 simplrr 538 . . . . . . . . 9 |- ( ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) /\ DECID x e. u ) -> v e. ( S \ u ) )
2120eldifad 3225 . . . . . . . 8 |- ( ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) /\ DECID x e. u ) -> v e. S )
2219, 21sseldd 3243 . . . . . . 7 |- ( ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) /\ DECID x e. u ) -> v e. A )
23 fidceq 7137 . . . . . . 7 |- ( ( A e. Fin /\ x e. A /\ v e. A ) -> DECID x = v )
2416, 17, 22, 23syl3anc 1274 . . . . . 6 |- ( ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) /\ DECID x e. u ) -> DECID x = v )
25 velsn 3711 . . . . . . 7 |- ( x e. { v } <-> x = v )
2625dcbii 848 . . . . . 6 |- (DECID x e. { v } <-> DECID x = v )
2724, 26sylibr 134 . . . . 5 |- ( ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) /\ DECID x e. u ) -> DECID x e. { v } )
2814, 27dcun 3623 . . . 4 |- ( ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) /\ DECID x e. u ) -> DECID x e. ( u u. { v } ) )
2928ex 115 . . 3 |- ( ( ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) /\ u e. Fin ) /\ ( u C_ S /\ v e. ( S \ u ) ) ) -> (DECID x e. u -> DECID x e. ( u u. { v } ) ) )
30 simpl3 1029 . . 3 |- ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) -> S e. Fin )
312, 4, 6, 8, 13, 29, 30findcard2sd 7162 . 2 |- ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) -> DECID x e. S )
3231ralrimiva 2617 1 |- ( ( S C_ A /\ A e. Fin /\ S e. Fin ) -> A. x e. A DECID x e. S )
Colors of variables: wff set class
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   \ cdif 3211   u. cun 3212   C_ wss 3214   (/)c0 3512   {csn 3694   Fincfn 6988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-er 6780  df-en 6989  df-fin 6991
This theorem is referenced by:  2omapfi  7284  hashfibclem  11231
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