Theorem fiss 7175

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 4308	. . . . 5 |- ( A C_ B <-> ~P A C_ ~P B )
3		ssrin 3432	. . . . 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 3292	. . . 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 2805	. . . 4 |- r e. _V
8		simpl 109	. . . . 5 |- ( ( B e. V /\ A C_ B ) -> B e. V )
9	8, 1	ssexd 4229	. . . 4 |- ( ( B e. V /\ A C_ B ) -> A e. _V )
10		elfi 7169	. . . 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 7169	. . . . 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 3233	1 |- ( ( B e. V /\ A C_ B ) -> ( fi ` A ) C_ ( fi ` B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   E.wrex 2511   _Vcvv 2802   i^i cin 3199   C_ wss 3200   ~Pcpw 3652   |^|cint 3928   ` cfv 5326   Fincfn 6908   ficfi 7166

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-er 6701  df-en 6909  df-fin 6911  df-fi 7167

This theorem is referenced by: (None)