Theorem elfir 6869

Description: Sufficient condition for an element of ( fi ` B ). (Contributed by Mario Carneiro, 24-Nov-2013.)

Assertion

Ref	Expression
elfir	|- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> |^| A e. ( fi ` B ) )

Proof of Theorem elfir

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

Step	Hyp	Ref	Expression
1		simp1 982	. . . . . 6 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> A C_ B )
2		elpw2g 4089	. . . . . 6 |- ( B e. V -> ( A e. ~P B <-> A C_ B ) )
3	1, 2	syl5ibr 155	. . . . 5 |- ( B e. V -> ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> A e. ~P B ) )
4	3	imp 123	. . . 4 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. ~P B )
5		simpr3 990	. . . 4 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. Fin )
6	4, 5	elind 3266	. . 3 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. ( ~P B i^i Fin ) )
7		eqid 2140	. . 3 |- |^| A = |^| A
8		inteq 3782	. . . 4 |- ( x = A -> |^| x = |^| A )
9	8	rspceeqv 2811	. . 3 |- ( ( A e. ( ~P B i^i Fin ) /\ |^| A = |^| A ) -> E. x e. ( ~P B i^i Fin ) |^| A = |^| x )
10	6, 7, 9	sylancl 410	. 2 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> E. x e. ( ~P B i^i Fin ) |^| A = |^| x )
11		simp2 983	. . . . 5 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> A =/= (/) )
12		fin0 6787	. . . . . 6 |- ( A e. Fin -> ( A =/= (/) <-> E. z z e. A ) )
13	12	3ad2ant3 1005	. . . . 5 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> ( A =/= (/) <-> E. z z e. A ) )
14	11, 13	mpbid 146	. . . 4 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> E. z z e. A )
15		inteximm 4082	. . . 4 |- ( E. z z e. A -> |^| A e. _V )
16	14, 15	syl 14	. . 3 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> |^| A e. _V )
17		id 19	. . 3 |- ( B e. V -> B e. V )
18		elfi 6867	. . 3 |- ( ( |^| A e. _V /\ B e. V ) -> ( |^| A e. ( fi ` B ) <-> E. x e. ( ~P B i^i Fin ) |^| A = |^| x ) )
19	16, 17, 18	syl2anr 288	. 2 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> ( |^| A e. ( fi ` B ) <-> E. x e. ( ~P B i^i Fin ) |^| A = |^| x ) )
20	10, 19	mpbird 166	1 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> |^| A e. ( fi ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1332   E.wex 1469   e. wcel 1481   =/= wne 2309   E.wrex 2418   _Vcvv 2689   i^i cin 3075   C_ wss 3076   (/)c0 3368   ~Pcpw 3515   |^|cint 3779   ` cfv 5131   Fincfn 6642   ficfi 6864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-er 6437  df-en 6643  df-fin 6645  df-fi 6865

This theorem is referenced by:  ssfii  6870