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Theorem fissfi 7256
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 2302 . . . 4 |- ( w = (/) -> ( x e. w <-> x e. (/) ) )
21dcbid 850 . . 3 |- ( w = (/) -> (DECID x e. w <-> DECID x e. (/) ) )
3 eleq2 2302 . . . 4 |- ( w = u -> ( x e. w <-> x e. u ) )
43dcbid 850 . . 3 |- ( w = u -> (DECID x e. w <-> DECID x e. u ) )
5 eleq2 2302 . . . 4 |- ( w = ( u u. { v } ) -> ( x e. w <-> x e. ( u u. { v } ) ) )
65dcbid 850 . . 3 |- ( w = ( u u. { v } ) -> (DECID x e. w <-> DECID x e. ( u u. { v } ) ) )
7 eleq2 2302 . . . 4 |- ( w = S -> ( x e. w <-> x e. S ) )
87dcbid 850 . . 3 |- ( w = S -> (DECID x e. w <-> DECID x e. S ) )
9 noel 3525 . . . . . 6 |- -. x e. (/)
109olci 744 . . . . 5 |- ( x e. (/) \/ -. x e. (/) )
11 df-dc 847 . . . . 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 1029 . . . . . . . 8 |- ( ( S C_ A /\ A e. Fin /\ S e. Fin ) -> A e. Fin )
1615ad4antr 498 . . . . . . 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 548 . . . . . . 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 1028 . . . . . . . . 9 |- ( ( S C_ A /\ A e. Fin /\ S e. Fin ) -> S C_ A )
1918ad4antr 498 . . . . . . . 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 542 . . . . . . . . 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 3231 . . . . . . . 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 3249 . . . . . . 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 7164 . . . . . . 7 |- ( ( A e. Fin /\ x e. A /\ v e. A ) -> DECID x = v )
2416, 17, 22, 23syl3anc 1278 . . . . . 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 3725 . . . . . . 7 |- ( x e. { v } <-> x = v )
2625dcbii 852 . . . . . 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 3637 . . . 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 1033 . . 3 |- ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) -> S e. Fin )
312, 4, 6, 8, 13, 29, 30findcard2sd 7189 . 2 |- ( ( ( S C_ A /\ A e. Fin /\ S e. Fin ) /\ x e. A ) -> DECID x e. S )
3231ralrimiva 2623 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 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   \ cdif 3217   u. cun 3218   C_ wss 3220   (/)c0 3520   {csn 3708   Fincfn 7015
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-iord 4509  df-on 4511  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-er 6800  df-en 7016  df-fin 7018
This theorem is referenced by:  2omapfi  7313  hashfibclem  11263
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