Theorem elfir 6938

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 987	. . . . . 6 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> A C_ B )
2		elpw2g 4135	. . . . . 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 995	. . . 4 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. Fin )
6	4, 5	elind 3307	. . 3 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. ( ~P B i^i Fin ) )
7		eqid 2165	. . 3 |- |^| A = |^| A
8		inteq 3827	. . . 4 |- ( x = A -> |^| x = |^| A )
9	8	rspceeqv 2848	. . 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 988	. . . . 5 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> A =/= (/) )
12		fin0 6851	. . . . . 6 |- ( A e. Fin -> ( A =/= (/) <-> E. z z e. A ) )
13	12	3ad2ant3 1010	. . . . 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 4128	. . . 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 6936	. . 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 968   = wceq 1343   E.wex 1480   e. wcel 2136   =/= wne 2336   E.wrex 2445   _Vcvv 2726   i^i cin 3115   C_ wss 3116   (/)c0 3409   ~Pcpw 3559   |^|cint 3824   ` cfv 5188   Fincfn 6706   ficfi 6933

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-er 6501  df-en 6707  df-fin 6709  df-fi 6934

This theorem is referenced by:  ssfii  6939