Theorem fiss 7043

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 4249	. . . . 5 |- ( A C_ B <-> ~P A C_ ~P B )
3		ssrin 3388	. . . . 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 3248	. . . 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 2766	. . . 4 |- r e. _V
8		simpl 109	. . . . 5 |- ( ( B e. V /\ A C_ B ) -> B e. V )
9	8, 1	ssexd 4173	. . . 4 |- ( ( B e. V /\ A C_ B ) -> A e. _V )
10		elfi 7037	. . . 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 7037	. . . . 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 3189	1 |- ( ( B e. V /\ A C_ B ) -> ( fi ` A ) C_ ( fi ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2167   E.wrex 2476   _Vcvv 2763   i^i cin 3156   C_ wss 3157   ~Pcpw 3605   |^|cint 3874   ` cfv 5258   Fincfn 6799   ficfi 7034

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-er 6592  df-en 6800  df-fin 6802  df-fi 7035

This theorem is referenced by: (None)