Theorem fliftf 6565

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 477	. . . . 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 6559	. . . . . . . . . 10 |- ( ph -> ( y F z <-> E. x e. X ( y = A /\ z = B ) ) )
6	5	exbidv 1850	. . . . . . . . 9 |- ( ph -> ( E. z y F z <-> E. z E. x e. X ( y = A /\ z = B ) ) )
7	6	adantr 481	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( E. z y F z <-> E. z E. x e. X ( y = A /\ z = B ) ) )
8		rexcom4 3225	. . . . . . . . 9 |- ( E. x e. X E. z ( y = A /\ z = B ) <-> E. z E. x e. X ( y = A /\ z = B ) )
9		elisset 3215	. . . . . . . . . . . . . 14 |- ( B e. S -> E. z z = B )
10	4, 9	syl 17	. . . . . . . . . . . . 13 |- ( ( ph /\ x e. X ) -> E. z z = B )
11	10	biantrud 528	. . . . . . . . . . . 12 |- ( ( ph /\ x e. X ) -> ( y = A <-> ( y = A /\ E. z z = B ) ) )
12		19.42v 1918	. . . . . . . . . . . 12 |- ( E. z ( y = A /\ z = B ) <-> ( y = A /\ E. z z = B ) )
13	11, 12	syl6rbbr 279	. . . . . . . . . . 11 |- ( ( ph /\ x e. X ) -> ( E. z ( y = A /\ z = B ) <-> y = A ) )
14	13	rexbidva 3049	. . . . . . . . . 10 |- ( ph -> ( E. x e. X E. z ( y = A /\ z = B ) <-> E. x e. X y = A ) )
15	14	adantr 481	. . . . . . . . 9 |- ( ( ph /\ Fun F ) -> ( E. x e. X E. z ( y = A /\ z = B ) <-> E. x e. X y = A ) )
16	8, 15	syl5bbr 274	. . . . . . . 8 |- ( ( ph /\ Fun F ) -> ( E. z E. x e. X ( y = A /\ z = B ) <-> E. x e. X y = A ) )
17	7, 16	bitrd 268	. . . . . . 7 |- ( ( ph /\ Fun F ) -> ( E. z y F z <-> E. x e. X y = A ) )
18	17	abbidv 2741	. . . . . 6 |- ( ( ph /\ Fun F ) -> { y | E. z y F z } = { y | E. x e. X y = A } )
19		df-dm 5124	. . . . . 6 |- dom F = { y | E. z y F z }
20		eqid 2622	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
21	20	rnmpt 5371	. . . . . 6 |- ran ( x e. X |-> A ) = { y | E. x e. X y = A }
22	18, 19, 21	3eqtr4g 2681	. . . . 5 |- ( ( ph /\ Fun F ) -> dom F = ran ( x e. X |-> A ) )
23		df-fn 5891	. . . . 5 |- ( F Fn ran ( x e. X |-> A ) <-> ( Fun F /\ dom F = ran ( x e. X |-> A ) ) )
24	1, 22, 23	sylanbrc 698	. . . 4 |- ( ( ph /\ Fun F ) -> F Fn ran ( x e. X |-> A ) )
25	2, 3, 4	fliftrel 6558	. . . . . . 7 |- ( ph -> F C_ ( R X. S ) )
26	25	adantr 481	. . . . . 6 |- ( ( ph /\ Fun F ) -> F C_ ( R X. S ) )
27		rnss 5354	. . . . . 6 |- ( F C_ ( R X. S ) -> ran F C_ ran ( R X. S ) )
28	26, 27	syl 17	. . . . 5 |- ( ( ph /\ Fun F ) -> ran F C_ ran ( R X. S ) )
29		rnxpss 5566	. . . . 5 |- ran ( R X. S ) C_ S
30	28, 29	syl6ss 3615	. . . 4 |- ( ( ph /\ Fun F ) -> ran F C_ S )
31		df-f 5892	. . . 4 |- ( F : ran ( x e. X |-> A ) --> S <-> ( F Fn ran ( x e. X |-> A ) /\ ran F C_ S ) )
32	24, 30, 31	sylanbrc 698	. . 3 |- ( ( ph /\ Fun F ) -> F : ran ( x e. X |-> A ) --> S )
33	32	ex 450	. 2 |- ( ph -> ( Fun F -> F : ran ( x e. X |-> A ) --> S ) )
34		ffun 6048	. 2 |- ( F : ran ( x e. X |-> A ) --> S -> Fun F )
35	33, 34	impbid1 215	1 |- ( ph -> ( Fun F <-> F : ran ( x e. X |-> A ) --> S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   E.wrex 2913   C_ wss 3574   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   -->wf 5884

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-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-fv 5896

This theorem is referenced by:  qliftf  7835  cygznlem2a  19916  pi1xfrf  22853  pi1cof  22859