Theorem fliftf 5459

Description: The domain and range of the function F. (Contributed by Mario Carneiro, 23-Dec-2016.)

Hypotheses

Ref	Expression
flift.1	|- F = ran ( x e. X |-> <. A , B >. )
flift.2	|- ( ( ph /\ x e. X ) -> A e. R )
flift.3	|- ( ( ph /\ x e. X ) -> B e. S )

Assertion

Ref	Expression
fliftf	|- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) )

Distinct variable groups:   x, R   ph, x   x, X   x, S

Allowed substitution hints:   A( x)   B( x)   F( x)

Proof of Theorem fliftf

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

Step	Hyp	Ref	Expression
1		simpr 108	. . . . 5 |- ( ( ph /\ Fun F ) -> Fun F )
2		flift.1	. . . . . . . . . . 11 |- F = ran ( x e. X |-> <. A , B >. )
3		flift.2	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> A e. R )
4		flift.3	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> B e. S )
5	2, 3, 4	fliftel 5453	. . . . . . . . . 10 |- ( ph -> ( y F z <-> E. x e. X ( y = A /\ z = B ) ) )
6	5	exbidv 1746	. . . . . . . . 9 |- ( ph -> ( E. z y F z <-> E. z E. x e. X ( y = A /\ z = B ) ) )
7	6	adantr 270	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( E. z y F z <-> E. z E. x e. X ( y = A /\ z = B ) ) )
8		rexcom4 2622	. . . . . . . . 9 |- ( E. x e. X E. z ( y = A /\ z = B ) <-> E. z E. x e. X ( y = A /\ z = B ) )
9		elisset 2613	. . . . . . . . . . . . . 14 |- ( B e. S -> E. z z = B )
10	4, 9	syl 14	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. X ) -> E. z z = B )
11	10	biantrud 298	. . . . . . . . . . . 12 |- ( ( ph /\ x e. X ) -> ( y = A <-> ( y = A /\ E. z z = B ) ) )
12		19.42v 1827	. . . . . . . . . . . 12 |- ( E. z ( y = A /\ z = B ) <-> ( y = A /\ E. z z = B ) )
13	11, 12	syl6rbbr 197	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( E. z ( y = A /\ z = B ) <-> y = A ) )
14	13	rexbidva 2365	. . . . . . . . . 10 |- ( ph -> ( E. x e. X E. z ( y = A /\ z = B ) <-> E. x e. X y = A ) )
15	14	adantr 270	. . . . . . . . 9 |- ( ( ph /\ Fun F ) -> ( E. x e. X E. z ( y = A /\ z = B ) <-> E. x e. X y = A ) )
16	8, 15	syl5bbr 192	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( E. z E. x e. X ( y = A /\ z = B ) <-> E. x e. X y = A ) )
17	7, 16	bitrd 186	. . . . . . 7 |- ( ( ph /\ Fun F ) -> ( E. z y F z <-> E. x e. X y = A ) )
18	17	abbidv 2196	. . . . . 6 |- ( ( ph /\ Fun F ) -> { y | E. z y F z } = { y | E. x e. X y = A } )
19		df-dm 4373	. . . . . 6 |- dom F = { y | E. z y F z }
20		eqid 2081	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
21	20	rnmpt 4600	. . . . . 6 |- ran ( x e. X |-> A ) = { y | E. x e. X y = A }
22	18, 19, 21	3eqtr4g 2138	. . . . 5 |- ( ( ph /\ Fun F ) -> dom F = ran ( x e. X |-> A ) )
23		df-fn 4925	. . . . 5 |- ( F Fn ran ( x e. X |-> A ) <-> ( Fun F /\ dom F = ran ( x e. X |-> A ) ) )
24	1, 22, 23	sylanbrc 408	. . . 4 |- ( ( ph /\ Fun F ) -> F Fn ran ( x e. X |-> A ) )
25	2, 3, 4	fliftrel 5452	. . . . . . 7 |- ( ph -> F C_ ( R X. S ) )
26	25	adantr 270	. . . . . 6 |- ( ( ph /\ Fun F ) -> F C_ ( R X. S ) )
27		rnss 4582	. . . . . 6 |- ( F C_ ( R X. S ) -> ran F C_ ran ( R X. S ) )
28	26, 27	syl 14	. . . . 5 |- ( ( ph /\ Fun F ) -> ran F C_ ran ( R X. S ) )
29		rnxpss 4774	. . . . 5 |- ran ( R X. S ) C_ S
30	28, 29	syl6ss 3011	. . . 4 |- ( ( ph /\ Fun F ) -> ran F C_ S )
31		df-f 4926	. . . 4 |- ( F : ran ( x e. X |-> A ) --> S <-> ( F Fn ran ( x e. X |-> A ) /\ ran F C_ S ) )
32	24, 30, 31	sylanbrc 408	. . 3 |- ( ( ph /\ Fun F ) -> F : ran ( x e. X |-> A ) --> S )
33	32	ex 113	. 2 |- ( ph -> ( Fun F -> F : ran ( x e. X |-> A ) --> S ) )
34		ffun 5068	. 2 |- ( F : ran ( x e. X |-> A ) --> S -> Fun F )
35	33, 34	impbid1 140	1 |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   E.wrex 2349   C_ wss 2973   <.cop 3401   class class class wbr 3785   |-> cmpt 3839   X. cxp 4361   dom cdm 4363   ran crn 4364   Fun wfun 4916   Fn wfn 4917   -->wf 4918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930

This theorem is referenced by:  qliftf  6214