Theorem relresfin 7268

Description: Restricting to a subset of a finite relation's domain gives a finite set. (Contributed by Jim Kingdon, 6-Oct-2026.)

Hypotheses

Ref	Expression
relresfin.e	|- ( ph -> E e. Fin )
relresfin.r	|- ( ph -> Rel E )
relresfin.dm	|- ( ph -> dom E e. Fin )
relresfin.ss	|- ( ph -> B C_ dom E )
relresfin.b	|- ( ph -> B e. Fin )

Assertion

Ref	Expression
relresfin	|- ( ph -> ( E |` B ) e. Fin )

Proof of Theorem relresfin

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relresfin.e	. 2 |- ( ph -> E e. Fin )
2		resss 5089	. . 3 |- ( E |` B ) C_ E
3	2	a1i 9	. 2 |- ( ph -> ( E |` B ) C_ E )
4		eleq1 2301	. . . . . 6 |- ( y = ( 1st ` x ) -> ( y e. B <-> ( 1st ` x ) e. B ) )
5	4	dcbid 850	. . . . 5 |- ( y = ( 1st ` x ) -> (DECID y e. B <-> DECID ( 1st ` x ) e. B ) )
6		relresfin.ss	. . . . . . 7 |- ( ph -> B C_ dom E )
7		relresfin.dm	. . . . . . 7 |- ( ph -> dom E e. Fin )
8		relresfin.b	. . . . . . 7 |- ( ph -> B e. Fin )
9		fissfi 7267	. . . . . . 7 |- ( ( B C_ dom E /\ dom E e. Fin /\ B e. Fin ) -> A. y e. dom EDECID y e. B )
10	6, 7, 8, 9	syl3anc 1278	. . . . . 6 |- ( ph -> A. y e. dom EDECID y e. B )
11	10	adantr 276	. . . . 5 |- ( ( ph /\ x e. E ) -> A. y e. dom EDECID y e. B )
12		simpr 110	. . . . . . 7 |- ( ( ph /\ x e. E ) -> x e. E )
13		relresfin.r	. . . . . . . . 9 |- ( ph -> Rel E )
14		1st2nd 6419	. . . . . . . . 9 |- ( ( Rel E /\ x e. E ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
15	13, 14	sylan 283	. . . . . . . 8 |- ( ( ph /\ x e. E ) -> x = <. ( 1st ` x ) , ( 2nd ` x ) >. )
16	15	eleq1d 2307	. . . . . . 7 |- ( ( ph /\ x e. E ) -> ( x e. E <-> <. ( 1st ` x ) , ( 2nd ` x ) >. e. E ) )
17	12, 16	mpbid 147	. . . . . 6 |- ( ( ph /\ x e. E ) -> <. ( 1st ` x ) , ( 2nd ` x ) >. e. E )
18		1stexg 6405	. . . . . . . 8 |- ( x e. _V -> ( 1st ` x ) e. _V )
19	18	elv 2825	. . . . . . 7 |- ( 1st ` x ) e. _V
20		2ndexg 6406	. . . . . . . 8 |- ( x e. _V -> ( 2nd ` x ) e. _V )
21	20	elv 2825	. . . . . . 7 |- ( 2nd ` x ) e. _V
22	19, 21	opeldm 4986	. . . . . 6 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. E -> ( 1st ` x ) e. dom E )
23	17, 22	syl 14	. . . . 5 |- ( ( ph /\ x e. E ) -> ( 1st ` x ) e. dom E )
24	5, 11, 23	rspcdva 2934	. . . 4 |- ( ( ph /\ x e. E ) -> DECID ( 1st ` x ) e. B )
25	17	biantrurd 305	. . . . . . 7 |- ( ( ph /\ x e. E ) -> ( ( 1st ` x ) e. B <-> ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. E /\ ( 1st ` x ) e. B ) ) )
26	21	opelres 5070	. . . . . . 7 |- ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( E |` B ) <-> ( <. ( 1st ` x ) , ( 2nd ` x ) >. e. E /\ ( 1st ` x ) e. B ) )
27	25, 26	bitr4di 198	. . . . . 6 |- ( ( ph /\ x e. E ) -> ( ( 1st ` x ) e. B <-> <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( E |` B ) ) )
28	15	eleq1d 2307	. . . . . 6 |- ( ( ph /\ x e. E ) -> ( x e. ( E |` B ) <-> <. ( 1st ` x ) , ( 2nd ` x ) >. e. ( E |` B ) ) )
29	27, 28	bitr4d 191	. . . . 5 |- ( ( ph /\ x e. E ) -> ( ( 1st ` x ) e. B <-> x e. ( E |` B ) ) )
30	29	dcbid 850	. . . 4 |- ( ( ph /\ x e. E ) -> (DECID ( 1st ` x ) e. B <-> DECID x e. ( E |` B ) ) )
31	24, 30	mpbid 147	. . 3 |- ( ( ph /\ x e. E ) -> DECID x e. ( E |` B ) )
32	31	ralrimiva 2623	. 2 |- ( ph -> A. x e. E DECID x e. ( E |` B ) )
33		ssfidc 7249	. 2 |- ( ( E e. Fin /\ ( E |` B ) C_ E /\ A. x e. E DECID x e. ( E |` B ) ) -> ( E |` B ) e. Fin )
34	1, 3, 32, 33	syl3anc 1278	1 |- ( ph -> ( E |` B ) e. Fin )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   <.cop 3712   dom cdm 4776   |` cres 4778   Rel wrel 4781   ` cfv 5380   1stc1st 6376   2ndc2nd 6377   Fincfn 7026

This proof depends on 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 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof 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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-1st 6378  df-2nd 6379  df-1o 6691  df-er 6811  df-en 7027  df-fin 7029

This theorem is used by:  gsum2dfilem2  14272