Theorem elfir 6910

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 4117	. . . . . 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 3292	. . 3 |- ( ( B e. V /\ ( A C_ B /\ A =/= (/) /\ A e. Fin ) ) -> A e. ( ~P B i^i Fin ) )
7		eqid 2157	. . 3 |- |^| A = |^| A
8		inteq 3810	. . . 4 |- ( x = A -> |^| x = |^| A )
9	8	rspceeqv 2834	. . 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 6823	. . . . . 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 4110	. . . 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 6908	. . 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 1335   E.wex 1472   e. wcel 2128   =/= wne 2327   E.wrex 2436   _Vcvv 2712   i^i cin 3101   C_ wss 3102   (/)c0 3394   ~Pcpw 3543   |^|cint 3807   ` cfv 5167   Fincfn 6678   ficfi 6905

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4134  ax-pr 4168  ax-un 4392  ax-iinf 4545

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4252  df-suc 4330  df-iom 4548  df-xp 4589  df-rel 4590  df-cnv 4591  df-co 4592  df-dm 4593  df-rn 4594  df-res 4595  df-ima 4596  df-iota 5132  df-fun 5169  df-fn 5170  df-f 5171  df-f1 5172  df-fo 5173  df-f1o 5174  df-fv 5175  df-er 6473  df-en 6679  df-fin 6681  df-fi 6906

This theorem is referenced by:  ssfii  6911