Theorem elfir 7307

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 1028	. . . . . 6 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> A C_ B )
2		elpw2g 4292	. . . . . 6 |- ( B e. V -> ( A e. ~P B <-> A C_ B ) )
3	1, 2	imbitrrid 156	. . . . 5 |- ( B e. V -> ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> A e. ~P B ) )
4	3	imp 124	. . . 4 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. ~P B )
5		simpr3 1036	. . . 4 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. Fin )
6	4, 5	elind 3414	. . 3 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. ( ~P B i^i Fin ) )
7		eqid 2238	. . 3 |- |^| A = |^| A
8		inteq 3973	. . . 4 |- ( x = A -> |^| x = |^| A )
9	8	rspceeqv 2948	. . 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 417	. 2 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> E. x e. ( ~P B i^i Fin ) |^| A = |^| x )
11		simp2 1029	. . . . 5 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> A =/= (/) )
12		fin0 7189	. . . . . 6 |- ( A e. Fin -> ( A =/= (/) <-> E. z z e. A ) )
13	12	3ad2ant3 1051	. . . . 5 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> ( A =/= (/) <-> E. z z e. A ) )
14	11, 13	mpbid 147	. . . 4 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> E. z z e. A )
15		inteximm 4285	. . . 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 7305	. . 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 290	. 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 167	1 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> |^| A e. ( fi ` B ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   =/= wne 2420   E.wrex 2529   _Vcvv 2821   i^i cin 3219   C_ wss 3220   (/)c0 3520   ~Pcpw 3688   |^|cint 3970   ` cfv 5377   Fincfn 7022   ficfi 7302

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-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

This proof depends on definitions:  df-bi 117  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-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-er 6807  df-en 7023  df-fin 7025  df-fi 7303

This theorem is used by:  ssfii  7308