Theorem fmpt 5340

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 5045	. . 3 |- ( A. x e. A C e. B -> F Fn A )
3	1	rnmpt 4600	. . . 4 |- ran F = { y | E. x e. A y = C }
4		r19.29 2494	. . . . . . 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 2141	. . . . . . . . 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 2473	. . . . . . 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 3068	. . . 4 |- ( A. x e. A C e. B -> { y | E. x e. A y = C } C_ B )
11	3, 10	syl5eqss 3043	. . 3 |- ( A. x e. A C e. B -> ran F C_ B )
12		df-f 4926	. . 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 4834	. . . 4 |- ( `' F " B ) = { x e. A | C e. B }
15		fimacnv 5317	. . . 4 |- ( F : A --> B -> ( `' F " B ) = A )
16	14, 15	syl5reqr 2128	. . 3 |- ( F : A --> B -> A = { x e. A | C e. B } )
17		rabid2 2530	. . 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 1284   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   {crab 2352   C_ wss 2973   |-> cmpt 3839   `'ccnv 4362   ran crn 4364   "cima 4366   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:  f1ompt  5341  fmpti  5342  fmptd  5343  rnmptss  5347  f1oresrab  5350  idref  5417  f1mpt  5431  f1stres  5806  f2ndres  5807  fmpt2x  5846  fmpt2co  5857  iunon  5922  dom2lem  6275  uzf  8622