Theorem intrnfi 8322

Description: Sufficient condition for the intersection of the range of a function to be in the set of finite intersections. (Contributed by Mario Carneiro, 30-Aug-2015.)

Assertion

Ref	Expression
intrnfi	|- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> |^| ran F e. ( fi ` B ) )

Proof of Theorem intrnfi

Step	Hyp	Ref	Expression
1		simpr1 1067	. . . 4 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> F : A --> B )
2		frn 6053	. . . 4 |- ( F : A --> B -> ran F C_ B )
3	1, 2	syl 17	. . 3 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> ran F C_ B )
4		fdm 6051	. . . . . 6 |- ( F : A --> B -> dom F = A )
5	1, 4	syl 17	. . . . 5 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> dom F = A )
6		simpr2 1068	. . . . 5 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> A =/= (/) )
7	5, 6	eqnetrd 2861	. . . 4 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> dom F =/= (/) )
8		dm0rn0 5342	. . . . 5 |- ( dom F = (/) <-> ran F = (/) )
9	8	necon3bii 2846	. . . 4 |- ( dom F =/= (/) <-> ran F =/= (/) )
10	7, 9	sylib 208	. . 3 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> ran F =/= (/) )
11		simpr3 1069	. . . 4 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> A e. Fin )
12		ffn 6045	. . . . . 6 |- ( F : A --> B -> F Fn A )
13	1, 12	syl 17	. . . . 5 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> F Fn A )
14		dffn4 6121	. . . . 5 |- ( F Fn A <-> F : A -onto-> ran F )
15	13, 14	sylib 208	. . . 4 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> F : A -onto-> ran F )
16		fofi 8252	. . . 4 |- ( ( A e. Fin /\ F : A -onto-> ran F ) -> ran F e. Fin )
17	11, 15, 16	syl2anc 693	. . 3 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> ran F e. Fin )
18	3, 10, 17	3jca 1242	. 2 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> ( ran F C_ B /\ ran F =/= (/) /\ ran F e. Fin ) )
19		elfir 8321	. 2 |- ( ( B e. V /\ ( ran F C_ B /\ ran F =/= (/) /\ ran F e. Fin ) ) -> |^| ran F e. ( fi ` B ) )
20	18, 19	syldan 487	1 |- ( ( B e. V /\ ( F : A --> B /\ A =/= (/) /\ A e. Fin ) ) -> |^| ran F e. ( fi ` B ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   |^|cint 4475   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888   Fincfn 7955   ficfi 8316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-fi 8317

This theorem is referenced by:  iinfi  8323  firest  16093