Theorem fmpt 5449

Description: Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)

Hypothesis

Ref	Expression
fmpt.1	|- F = ( x e. A |-> C )

Assertion

Ref	Expression
fmpt	|- ( A. x e. A C e. B <-> F : A --> B )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   C( x)   F( x)

Proof of Theorem fmpt

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmpt.1	. . . 4 |- F = ( x e. A |-> C )
2	1	fnmpt 5140	. . 3 |- ( A. x e. A C e. B -> F Fn A )
3	1	rnmpt 4683	. . . 4 |- ran F = { y | E. x e. A y = C }
4		r19.29 2506	. . . . . . 7 |- ( ( A. x e. A C e. B /\ E. x e. A y = C ) -> E. x e. A ( C e. B /\ y = C ) )
5		eleq1 2150	. . . . . . . . 9 |- ( y = C -> ( y e. B <-> C e. B ) )
6	5	biimparc 293	. . . . . . . 8 |- ( ( C e. B /\ y = C ) -> y e. B )
7	6	rexlimivw 2485	. . . . . . 7 |- ( E. x e. A ( C e. B /\ y = C ) -> y e. B )
8	4, 7	syl 14	. . . . . 6 |- ( ( A. x e. A C e. B /\ E. x e. A y = C ) -> y e. B )
9	8	ex 113	. . . . 5 |- ( A. x e. A C e. B -> ( E. x e. A y = C -> y e. B ) )
10	9	abssdv 3095	. . . 4 |- ( A. x e. A C e. B -> { y | E. x e. A y = C } C_ B )
11	3, 10	syl5eqss 3070	. . 3 |- ( A. x e. A C e. B -> ran F C_ B )
12		df-f 5019	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
13	2, 11, 12	sylanbrc 408	. 2 |- ( A. x e. A C e. B -> F : A --> B )
14	1	mptpreima 4924	. . . 4 |- ( `' F " B ) = { x e. A | C e. B }
15		fimacnv 5428	. . . 4 |- ( F : A --> B -> ( `' F " B ) = A )
16	14, 15	syl5reqr 2135	. . 3 |- ( F : A --> B -> A = { x e. A | C e. B } )
17		rabid2 2543	. . 3 |- ( A = { x e. A | C e. B } <-> A. x e. A C e. B )
18	16, 17	sylib 120	. 2 |- ( F : A --> B -> A. x e. A C e. B )
19	13, 18	impbii 124	1 |- ( A. x e. A C e. B <-> F : A --> B )

Syntax hints:   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   {crab 2363   C_ wss 2999   |-> cmpt 3899   `'ccnv 4437   ran crn 4439   "cima 4441   Fn wfn 5010   -->wf 5011

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023

This theorem is referenced by:  f1ompt  5450  fmpti  5451  fmptd  5452  fmptdf  5455  rnmptss  5459  f1oresrab  5463  idref  5536  f1mpt  5550  f1stres  5930  f2ndres  5931  fmpt2x  5970  fmpt2co  5981  iunon  6049  dom2lem  6487  uzf  9020