Theorem elfir 7034

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 999	. . . . . 6 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> A C_ B )
2		elpw2g 4186	. . . . . 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 1007	. . . 4 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. Fin )
6	4, 5	elind 3345	. . 3 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. ( ~P B i^i Fin ) )
7		eqid 2193	. . 3 |- |^| A = |^| A
8		inteq 3874	. . . 4 |- ( x = A -> |^| x = |^| A )
9	8	rspceeqv 2883	. . 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 413	. 2 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> E. x e. ( ~P B i^i Fin ) |^| A = |^| x )
11		simp2 1000	. . . . 5 |- ( ( A C_ B /\ A =/= (/) /\ A e. Fin ) -> A =/= (/) )
12		fin0 6943	. . . . . 6 |- ( A e. Fin -> ( A =/= (/) <-> E. z z e. A ) )
13	12	3ad2ant3 1022	. . . . 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 4179	. . . 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 7032	. . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   E.wex 1503   e. wcel 2164   =/= wne 2364   E.wrex 2473   _Vcvv 2760   i^i cin 3153   C_ wss 3154   (/)c0 3447   ~Pcpw 3602   |^|cint 3871   ` cfv 5255   Fincfn 6796   ficfi 7029

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6589  df-en 6797  df-fin 6799  df-fi 7030

This theorem is referenced by:  ssfii  7035