Theorem fmpt 5319

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 5025	. . 3 |- ( A. x e. A C e. B -> F Fn A )
3	1	rnmpt 4582	. . . 4 |- ran F = { y | E. x e. A y = C }
4		r19.29 2450	. . . . . . 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 2100	. . . . . . . . 9 |- ( y = C -> ( y e. B <-> C e. B ) )
6	5	biimparc 283	. . . . . . . 8 |- ( ( C e. B /\ y = C ) -> y e. B )
7	6	rexlimivw 2429	. . . . . . 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 108	. . . . 5 |- ( A. x e. A C e. B -> ( E. x e. A y = C -> y e. B ) )
10	9	abssdv 3014	. . . 4 |- ( A. x e. A C e. B -> { y | E. x e. A y = C } C_ B )
11	3, 10	syl5eqss 2989	. . 3 |- ( A. x e. A C e. B -> ran F C_ B )
12		df-f 4906	. . 3 |- ( F : A --> B <-> ( F Fn A /\ ran F C_ B ) )
13	2, 11, 12	sylanbrc 394	. 2 |- ( A. x e. A C e. B -> F : A --> B )
14	1	mptpreima 4814	. . . 4 |- ( `' F " B ) = { x e. A | C e. B }
15		fimacnv 5296	. . . 4 |- ( F : A --> B -> ( `' F " B ) = A )
16	14, 15	syl5reqr 2087	. . 3 |- ( F : A --> B -> A = { x e. A | C e. B } )
17		rabid2 2486	. . 3 |- ( A = { x e. A | C e. B } <-> A. x e. A C e. B )
18	16, 17	sylib 127	. 2 |- ( F : A --> B -> A. x e. A C e. B )
19	13, 18	impbii 117	1 |- ( A. x e. A C e. B <-> F : A --> B )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   {crab 2310   C_ wss 2917   |-> cmpt 3818   `'ccnv 4344   ran crn 4346   "cima 4348   Fn wfn 4897   -->wf 4898

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910

This theorem is referenced by:  f1ompt  5320  fmpti  5321  fmptd  5322  rnmptss  5326  f1oresrab  5329  idref  5396  f1mpt  5410  f1stres  5786  f2ndres  5787  fmpt2x  5826  fmpt2co  5837  iunon  5899  dom2lem  6252  uzf  8476