Theorem fiss 6873

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 4146	. . . . 5 |- ( A C_ B <-> ~P A C_ ~P B )
3		ssrin 3306	. . . . 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 3167	. . . 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 2692	. . . 4 |- r e. _V
8		simpl 108	. . . . 5 |- ( ( B e. V /\ A C_ B ) -> B e. V )
9	8, 1	ssexd 4076	. . . 4 |- ( ( B e. V /\ A C_ B ) -> A e. _V )
10		elfi 6867	. . . 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 411	. . 3 |- ( ( B e. V /\ A C_ B ) -> ( r e. ( fi ` A ) <-> E. x e. ( ~P A i^i Fin ) r = |^| x ) )
12		elfi 6867	. . . . 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 421	. . . 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 3108	1 |- ( ( B e. V /\ A C_ B ) -> ( fi ` A ) C_ ( fi ` B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1332   e. wcel 1481   E.wrex 2418   _Vcvv 2689   i^i cin 3075   C_ wss 3076   ~Pcpw 3515   |^|cint 3779   ` cfv 5131   Fincfn 6642   ficfi 6864

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-er 6437  df-en 6643  df-fin 6645  df-fi 6865

This theorem is referenced by: (None)