Theorem fmpt 5488

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 5174	. . 3 |- ( A. x e. A C e. B -> F Fn A )
3	1	rnmpt 4715	. . . 4 |- ran F = { y | E. x e. A y = C }
4		r19.29 2520	. . . . . . 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 2157	. . . . . . . . 9 |- ( y = C -> ( y e. B <-> C e. B ) )
6	5	biimparc 294	. . . . . . . 8 |- ( ( C e. B /\ y = C ) -> y e. B )
7	6	rexlimivw 2498	. . . . . . 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 114	. . . . 5 |- ( A. x e. A C e. B -> ( E. x e. A y = C -> y e. B ) )
10	9	abssdv 3110	. . . 4 |- ( A. x e. A C e. B -> { y | E. x e. A y = C } C_ B )
11	3, 10	syl5eqss 3085	. . 3 |- ( A. x e. A C e. B -> ran F C_ B )
12		df-f 5053	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
13	2, 11, 12	sylanbrc 409	. 2 |- ( A. x e. A C e. B -> F : A --> B )
14	1	mptpreima 4958	. . . 4 |- ( `' F " B ) = { x e. A | C e. B }
15		fimacnv 5467	. . . 4 |- ( F : A --> B -> ( `' F " B ) = A )
16	14, 15	syl5reqr 2142	. . 3 |- ( F : A --> B -> A = { x e. A | C e. B } )
17		rabid2 2557	. . 3 |- ( A = { x e. A | C e. B } <-> A. x e. A C e. B )
18	16, 17	sylib 121	. 2 |- ( F : A --> B -> A. x e. A C e. B )
19	13, 18	impbii 125	1 |- ( A. x e. A C e. B <-> F : A --> B )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1296   e. wcel 1445   {cab 2081   A.wral 2370   E.wrex 2371   {crab 2374   C_ wss 3013   |-> cmpt 3921   `'ccnv 4466   ran crn 4468   "cima 4470   Fn wfn 5044   -->wf 5045

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-rab 2379  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057

This theorem is referenced by:  f1ompt  5489  fmpti  5490  fmptd  5491  fmptdf  5494  rnmptss  5498  f1oresrab  5502  idref  5574  f1mpt  5588  f1stres  5968  f2ndres  5969  fmpt2x  6008  fmpt2co  6019  iunon  6087  mptelixpg  6531  dom2lem  6569  uzf  9121  cnmpt11  12105  cncfmpt1f  12348  mulcncflem  12353  mulcncf  12354