Theorem ofrn2 29442

Description: The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)

Hypotheses

Ref	Expression
ofrn.1	|- ( ph -> F : A --> B )
ofrn.2	|- ( ph -> G : A --> B )
ofrn.3	|- ( ph -> .+ : ( B X. B ) --> C )
ofrn.4	|- ( ph -> A e. V )

Assertion

Ref	Expression
ofrn2	|- ( ph -> ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) )

Proof of Theorem ofrn2

Dummy variables x y z a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofrn.1	. . . . . . . 8 |- ( ph -> F : A --> B )
2		ffn 6045	. . . . . . . 8 |- ( F : A --> B -> F Fn A )
3	1, 2	syl 17	. . . . . . 7 |- ( ph -> F Fn A )
4	3	adantr 481	. . . . . 6 |- ( ( ph /\ ( a e. A /\ z = ( ( F ` a ) .+ ( G ` a ) ) ) ) -> F Fn A )
5		simprl 794	. . . . . 6 |- ( ( ph /\ ( a e. A /\ z = ( ( F ` a ) .+ ( G ` a ) ) ) ) -> a e. A )
6		fnfvelrn 6356	. . . . . 6 |- ( ( F Fn A /\ a e. A ) -> ( F ` a ) e. ran F )
7	4, 5, 6	syl2anc 693	. . . . 5 |- ( ( ph /\ ( a e. A /\ z = ( ( F ` a ) .+ ( G ` a ) ) ) ) -> ( F ` a ) e. ran F )
8		ofrn.2	. . . . . . . 8 |- ( ph -> G : A --> B )
9		ffn 6045	. . . . . . . 8 |- ( G : A --> B -> G Fn A )
10	8, 9	syl 17	. . . . . . 7 |- ( ph -> G Fn A )
11	10	adantr 481	. . . . . 6 |- ( ( ph /\ ( a e. A /\ z = ( ( F ` a ) .+ ( G ` a ) ) ) ) -> G Fn A )
12		fnfvelrn 6356	. . . . . 6 |- ( ( G Fn A /\ a e. A ) -> ( G ` a ) e. ran G )
13	11, 5, 12	syl2anc 693	. . . . 5 |- ( ( ph /\ ( a e. A /\ z = ( ( F ` a ) .+ ( G ` a ) ) ) ) -> ( G ` a ) e. ran G )
14		simprr 796	. . . . 5 |- ( ( ph /\ ( a e. A /\ z = ( ( F ` a ) .+ ( G ` a ) ) ) ) -> z = ( ( F ` a ) .+ ( G ` a ) ) )
15		rspceov 6692	. . . . 5 |- ( ( ( F ` a ) e. ran F /\ ( G ` a ) e. ran G /\ z = ( ( F ` a ) .+ ( G ` a ) ) ) -> E. x e. ran F E. y e. ran G z = ( x .+ y ) )
16	7, 13, 14, 15	syl3anc 1326	. . . 4 |- ( ( ph /\ ( a e. A /\ z = ( ( F ` a ) .+ ( G ` a ) ) ) ) -> E. x e. ran F E. y e. ran G z = ( x .+ y ) )
17	16	rexlimdvaa 3032	. . 3 |- ( ph -> ( E. a e. A z = ( ( F ` a ) .+ ( G ` a ) ) -> E. x e. ran F E. y e. ran G z = ( x .+ y ) ) )
18	17	ss2abdv 3675	. 2 |- ( ph -> { z | E. a e. A z = ( ( F ` a ) .+ ( G ` a ) ) } C_ { z | E. x e. ran F E. y e. ran G z = ( x .+ y ) } )
19		ofrn.4	. . . . 5 |- ( ph -> A e. V )
20		inidm 3822	. . . . 5 |- ( A i^i A ) = A
21		eqidd 2623	. . . . 5 |- ( ( ph /\ a e. A ) -> ( F ` a ) = ( F ` a ) )
22		eqidd 2623	. . . . 5 |- ( ( ph /\ a e. A ) -> ( G ` a ) = ( G ` a ) )
23	3, 10, 19, 19, 20, 21, 22	offval 6904	. . . 4 |- ( ph -> ( F oF .+ G ) = ( a e. A |-> ( ( F ` a ) .+ ( G ` a ) ) ) )
24	23	rneqd 5353	. . 3 |- ( ph -> ran ( F oF .+ G ) = ran ( a e. A |-> ( ( F ` a ) .+ ( G ` a ) ) ) )
25		eqid 2622	. . . 4 |- ( a e. A |-> ( ( F ` a ) .+ ( G ` a ) ) ) = ( a e. A |-> ( ( F ` a ) .+ ( G ` a ) ) )
26	25	rnmpt 5371	. . 3 |- ran ( a e. A |-> ( ( F ` a ) .+ ( G ` a ) ) ) = { z | E. a e. A z = ( ( F ` a ) .+ ( G ` a ) ) }
27	24, 26	syl6eq 2672	. 2 |- ( ph -> ran ( F oF .+ G ) = { z | E. a e. A z = ( ( F ` a ) .+ ( G ` a ) ) } )
28		ofrn.3	. . . . 5 |- ( ph -> .+ : ( B X. B ) --> C )
29		ffn 6045	. . . . 5 |- ( .+ : ( B X. B ) --> C -> .+ Fn ( B X. B ) )
30	28, 29	syl 17	. . . 4 |- ( ph -> .+ Fn ( B X. B ) )
31		frn 6053	. . . . . 6 |- ( F : A --> B -> ran F C_ B )
32	1, 31	syl 17	. . . . 5 |- ( ph -> ran F C_ B )
33		frn 6053	. . . . . 6 |- ( G : A --> B -> ran G C_ B )
34	8, 33	syl 17	. . . . 5 |- ( ph -> ran G C_ B )
35		xpss12 5225	. . . . 5 |- ( ( ran F C_ B /\ ran G C_ B ) -> ( ran F X. ran G ) C_ ( B X. B ) )
36	32, 34, 35	syl2anc 693	. . . 4 |- ( ph -> ( ran F X. ran G ) C_ ( B X. B ) )
37		ovelimab 6812	. . . 4 |- ( ( .+ Fn ( B X. B ) /\ ( ran F X. ran G ) C_ ( B X. B ) ) -> ( z e. ( .+ " ( ran F X. ran G ) ) <-> E. x e. ran F E. y e. ran G z = ( x .+ y ) ) )
38	30, 36, 37	syl2anc 693	. . 3 |- ( ph -> ( z e. ( .+ " ( ran F X. ran G ) ) <-> E. x e. ran F E. y e. ran G z = ( x .+ y ) ) )
39	38	abbi2dv 2742	. 2 |- ( ph -> ( .+ " ( ran F X. ran G ) ) = { z | E. x e. ran F E. y e. ran G z = ( x .+ y ) } )
40	18, 27, 39	3sstr4d 3648	1 |- ( ph -> ran ( F oF .+ G ) C_ ( .+ " ( ran F X. ran G ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   |-> cmpt 4729   X. cxp 5112   ran crn 5115   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  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-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by:  sibfof  30402