Theorem funcoressn 41207

Description: A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)

Assertion

Ref	Expression
funcoressn	|- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> Fun ( ( F o. G ) |` { X } ) )

Proof of Theorem funcoressn

Step	Hyp	Ref	Expression
1		dmressnsn 5438	. . . . . . . 8 |- ( ( G ` X ) e. dom F -> dom ( F |` { ( G ` X ) } ) = { ( G ` X ) } )
2		df-fn 5891	. . . . . . . . 9 |- ( ( F |` { ( G ` X ) } ) Fn { ( G ` X ) } <-> ( Fun ( F |` { ( G ` X ) } ) /\ dom ( F |` { ( G ` X ) } ) = { ( G ` X ) } ) )
3	2	simplbi2com 657	. . . . . . . 8 |- ( dom ( F |` { ( G ` X ) } ) = { ( G ` X ) } -> ( Fun ( F |` { ( G ` X ) } ) -> ( F |` { ( G ` X ) } ) Fn { ( G ` X ) } ) )
4	1, 3	syl 17	. . . . . . 7 |- ( ( G ` X ) e. dom F -> ( Fun ( F |` { ( G ` X ) } ) -> ( F |` { ( G ` X ) } ) Fn { ( G ` X ) } ) )
5	4	imp 445	. . . . . 6 |- ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) -> ( F |` { ( G ` X ) } ) Fn { ( G ` X ) } )
6	5	adantr 481	. . . . 5 |- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> ( F |` { ( G ` X ) } ) Fn { ( G ` X ) } )
7		fnsnfv 6258	. . . . . . . . 9 |- ( ( G Fn A /\ X e. A ) -> { ( G ` X ) } = ( G " { X } ) )
8	7	adantl 482	. . . . . . . 8 |- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> { ( G ` X ) } = ( G " { X } ) )
9		df-ima 5127	. . . . . . . 8 |- ( G " { X } ) = ran ( G |` { X } )
10	8, 9	syl6eq 2672	. . . . . . 7 |- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> { ( G ` X ) } = ran ( G |` { X } ) )
11	10	reseq2d 5396	. . . . . 6 |- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> ( F |` { ( G ` X ) } ) = ( F |` ran ( G |` { X } ) ) )
12	11, 10	fneq12d 5983	. . . . 5 |- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> ( ( F |` { ( G ` X ) } ) Fn { ( G ` X ) } <-> ( F |` ran ( G |` { X } ) ) Fn ran ( G |` { X } ) ) )
13	6, 12	mpbid 222	. . . 4 |- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> ( F |` ran ( G |` { X } ) ) Fn ran ( G |` { X } ) )
14		fnfun 5988	. . . . . . 7 |- ( G Fn A -> Fun G )
15		funres 5929	. . . . . . . 8 |- ( Fun G -> Fun ( G |` { X } ) )
16		funfn 5918	. . . . . . . 8 |- ( Fun ( G |` { X } ) <-> ( G |` { X } ) Fn dom ( G |` { X } ) )
17	15, 16	sylib 208	. . . . . . 7 |- ( Fun G -> ( G |` { X } ) Fn dom ( G |` { X } ) )
18	14, 17	syl 17	. . . . . 6 |- ( G Fn A -> ( G |` { X } ) Fn dom ( G |` { X } ) )
19	18	adantr 481	. . . . 5 |- ( ( G Fn A /\ X e. A ) -> ( G |` { X } ) Fn dom ( G |` { X } ) )
20	19	adantl 482	. . . 4 |- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> ( G |` { X } ) Fn dom ( G |` { X } ) )
21		fnresfnco 41206	. . . 4 |- ( ( ( F |` ran ( G |` { X } ) ) Fn ran ( G |` { X } ) /\ ( G |` { X } ) Fn dom ( G |` { X } ) ) -> ( F o. ( G |` { X } ) ) Fn dom ( G |` { X } ) )
22	13, 20, 21	syl2anc 693	. . 3 |- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> ( F o. ( G |` { X } ) ) Fn dom ( G |` { X } ) )
23		fnfun 5988	. . 3 |- ( ( F o. ( G |` { X } ) ) Fn dom ( G |` { X } ) -> Fun ( F o. ( G |` { X } ) ) )
24	22, 23	syl 17	. 2 |- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> Fun ( F o. ( G |` { X } ) ) )
25		resco 5639	. . 3 |- ( ( F o. G ) |` { X } ) = ( F o. ( G |` { X } ) )
26	25	funeqi 5909	. 2 |- ( Fun ( ( F o. G ) |` { X } ) <-> Fun ( F o. ( G |` { X } ) ) )
27	24, 26	sylibr 224	1 |- ( ( ( ( G ` X ) e. dom F /\ Fun ( F |` { ( G ` X ) } ) ) /\ ( G Fn A /\ X e. A ) ) -> Fun ( ( F o. G ) |` { X } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {csn 4177   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   o. ccom 5118   Fun wfun 5882   Fn wfn 5883   ` cfv 5888

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  afvco2  41256