Theorem fiss 7236

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 110	. . . 4 |- ( ( B e. V /\ A C_ B ) -> A C_ B )
2		sspwb 4314	. . . . 5 |- ( A C_ B <-> ~P A C_ ~P B )
3		ssrin 3434	. . . . 5 |- ( ~P A C_ ~P B -> ( ~P A i^i Fin ) C_ ( ~P B i^i Fin ) )
4	2, 3	sylbi 121	. . . 4 |- ( A C_ B -> ( ~P A i^i Fin ) C_ ( ~P B i^i Fin ) )
5		ssrexv 3293	. . . 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 2806	. . . 4 |- r e. _V
8		simpl 109	. . . . 5 |- ( ( B e. V /\ A C_ B ) -> B e. V )
9	8, 1	ssexd 4234	. . . 4 |- ( ( B e. V /\ A C_ B ) -> A e. _V )
10		elfi 7230	. . . 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 414	. . 3 |- ( ( B e. V /\ A C_ B ) -> ( r e. ( fi ` A ) <-> E. x e. ( ~P A i^i Fin ) r = |^| x ) )
12		elfi 7230	. . . . 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 424	. . . 4 |- ( B e. V -> ( r e. ( fi ` B ) <-> E. x e. ( ~P B i^i Fin ) r = |^| x ) )
14	13	adantr 276	. . 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 203	. 2 |- ( ( B e. V /\ A C_ B ) -> ( r e. ( fi ` A ) -> r e. ( fi ` B ) ) )
16	15	ssrdv 3234	1 |- ( ( B e. V /\ A C_ B ) -> ( fi ` A ) C_ ( fi ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   E.wrex 2512   _Vcvv 2803   i^i cin 3200   C_ wss 3201   ~Pcpw 3656   |^|cint 3933   ` cfv 5333   Fincfn 6952   ficfi 7227

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-er 6745  df-en 6953  df-fin 6955  df-fi 7228

This theorem is referenced by: (None)