Theorem fiss 6865

Description: Subset relationship for function fi. (Contributed by Jeff Hankins, 7-Oct-2009.) (Revised by Mario Carneiro, 24-Nov-2013.)

Assertion

Ref	Expression
fiss	|- ( ( B e. V /\ A C_ B ) -> ( fi ` A ) C_ ( fi ` B ) )

Proof of Theorem fiss

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

Step	Hyp	Ref	Expression
1		simpr 109	. . . 4 |- ( ( B e. V /\ A C_ B ) -> A C_ B )
2		sspwb 4138	. . . . 5 |- ( A C_ B <-> ~P A C_ ~P B )
3		ssrin 3301	. . . . 5 |- ( ~P A C_ ~P B -> ( ~P A i^i Fin ) C_ ( ~P B i^i Fin ) )
4	2, 3	sylbi 120	. . . 4 |- ( A C_ B -> ( ~P A i^i Fin ) C_ ( ~P B i^i Fin ) )
5		ssrexv 3162	. . . 4 |- ( ( ~P A i^i Fin ) C_ ( ~P B i^i Fin ) -> ( E. x e. ( ~P A i^i Fin ) r = |^| x -> E. x e. ( ~P B i^i Fin ) r = |^| x ) )
6	1, 4, 5	3syl 17	. . 3 |- ( ( B e. V /\ A C_ B ) -> ( E. x e. ( ~P A i^i Fin ) r = |^| x -> E. x e. ( ~P B i^i Fin ) r = |^| x ) )
7		vex 2689	. . . 4 |- r e. _V
8		simpl 108	. . . . 5 |- ( ( B e. V /\ A C_ B ) -> B e. V )
9	8, 1	ssexd 4068	. . . 4 |- ( ( B e. V /\ A C_ B ) -> A e. _V )
10		elfi 6859	. . . 4 |- ( ( r e. _V /\ A e. _V ) -> ( r e. ( fi ` A ) <-> E. x e. ( ~P A i^i Fin ) r = |^| x ) )
11	7, 9, 10	sylancr 410	. . 3 |- ( ( B e. V /\ A C_ B ) -> ( r e. ( fi ` A ) <-> E. x e. ( ~P A i^i Fin ) r = |^| x ) )
12		elfi 6859	. . . . 5 |- ( ( r e. _V /\ B e. V ) -> ( r e. ( fi ` B ) <-> E. x e. ( ~P B i^i Fin ) r = |^| x ) )
13	7, 12	mpan 420	. . . 4 |- ( B e. V -> ( r e. ( fi ` B ) <-> E. x e. ( ~P B i^i Fin ) r = |^| x ) )
14	13	adantr 274	. . 3 |- ( ( B e. V /\ A C_ B ) -> ( r e. ( fi ` B ) <-> E. x e. ( ~P B i^i Fin ) r = |^| x ) )
15	6, 11, 14	3imtr4d 202	. 2 |- ( ( B e. V /\ A C_ B ) -> ( r e. ( fi ` A ) -> r e. ( fi ` B ) ) )
16	15	ssrdv 3103	1 |- ( ( B e. V /\ A C_ B ) -> ( fi ` A ) C_ ( fi ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2417   _Vcvv 2686   i^i cin 3070   C_ wss 3071   ~Pcpw 3510   |^|cint 3771   ` cfv 5123   Fincfn 6634   ficfi 6856

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-er 6429  df-en 6635  df-fin 6637  df-fi 6857

This theorem is referenced by: (None)