Theorem mptpreima 5483

Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Hypothesis

Ref	Expression
dmmpt2.1	|- F = ( x e. A |-> B )

Assertion

Ref	Expression
mptpreima	|- ( `' F " C ) = { x e. A | B e. C }

Distinct variable group:   x, C

Allowed substitution hints:   A( x)   B( x)   F( x)

Proof of Theorem mptpreima

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmmpt2.1	. . . . . 6 |- F = ( x e. A |-> B )
2		df-mpt 4499	. . . . . 6 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
3	1, 2	eqtri 2483	. . . . 5 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
4	3	cnveqi 5166	. . . 4 |- `' F = `' { <. x , y >. | ( x e. A /\ y = B ) }
5		cnvopab 5392	. . . 4 |- `' { <. x , y >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( x e. A /\ y = B ) }
6	4, 5	eqtri 2483	. . 3 |- `' F = { <. y , x >. | ( x e. A /\ y = B ) }
7	6	imaeq1i 5322	. 2 |- ( `' F " C ) = ( { <. y , x >. | ( x e. A /\ y = B ) } " C )
8		df-ima 5001	. . 3 |- ( { <. y , x >. | ( x e. A /\ y = B ) } " C ) = ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C )
9		resopab 5308	. . . . 5 |- ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) }
10	9	rneqi 5218	. . . 4 |- ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = ran { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) }
11		ancom 448	. . . . . . . . 9 |- ( ( y e. C /\ ( x e. A /\ y = B ) ) <-> ( ( x e. A /\ y = B ) /\ y e. C ) )
12		anass 647	. . . . . . . . 9 |- ( ( ( x e. A /\ y = B ) /\ y e. C ) <-> ( x e. A /\ ( y = B /\ y e. C ) ) )
13	11, 12	bitri 249	. . . . . . . 8 |- ( ( y e. C /\ ( x e. A /\ y = B ) ) <-> ( x e. A /\ ( y = B /\ y e. C ) ) )
14	13	exbii 1672	. . . . . . 7 |- ( E. y ( y e. C /\ ( x e. A /\ y = B ) ) <-> E. y ( x e. A /\ ( y = B /\ y e. C ) ) )
15		19.42v 1780	. . . . . . . 8 |- ( E. y ( x e. A /\ ( y = B /\ y e. C ) ) <-> ( x e. A /\ E. y ( y = B /\ y e. C ) ) )
16		df-clel 2449	. . . . . . . . . 10 |- ( B e. C <-> E. y ( y = B /\ y e. C ) )
17	16	bicomi 202	. . . . . . . . 9 |- ( E. y ( y = B /\ y e. C ) <-> B e. C )
18	17	anbi2i 692	. . . . . . . 8 |- ( ( x e. A /\ E. y ( y = B /\ y e. C ) ) <-> ( x e. A /\ B e. C ) )
19	15, 18	bitri 249	. . . . . . 7 |- ( E. y ( x e. A /\ ( y = B /\ y e. C ) ) <-> ( x e. A /\ B e. C ) )
20	14, 19	bitri 249	. . . . . 6 |- ( E. y ( y e. C /\ ( x e. A /\ y = B ) ) <-> ( x e. A /\ B e. C ) )
21	20	abbii 2588	. . . . 5 |- { x | E. y ( y e. C /\ ( x e. A /\ y = B ) ) } = { x | ( x e. A /\ B e. C ) }
22		rnopab 5236	. . . . 5 |- ran { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) } = { x | E. y ( y e. C /\ ( x e. A /\ y = B ) ) }
23		df-rab 2813	. . . . 5 |- { x e. A | B e. C } = { x | ( x e. A /\ B e. C ) }
24	21, 22, 23	3eqtr4i 2493	. . . 4 |- ran { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) } = { x e. A | B e. C }
25	10, 24	eqtri 2483	. . 3 |- ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = { x e. A | B e. C }
26	8, 25	eqtri 2483	. 2 |- ( { <. y , x >. | ( x e. A /\ y = B ) } " C ) = { x e. A | B e. C }
27	7, 26	eqtri 2483	1 |- ( `' F " C ) = { x e. A | B e. C }

Syntax hints:   /\ wa 367   = wceq 1398   E.wex 1617   e. wcel 1823   {cab 2439   {crab 2808   {copab 4496   |-> cmpt 4497   `'ccnv 4987   ran crn 4989   |` cres 4990   "cima 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-mpt 4499  df-xp 4994  df-rel 4995  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001

This theorem is referenced by:  mptiniseg  5484  dmmpt  5485  fmpt  6028  f1oresrab  6039  suppss2OLD  6503  suppssfvOLD  6504  suppssov1OLD  6505  mptsuppdifd  6914  cantnflem1OLD  8122  cantnfOLD  8125  r0weon  8381  compss  8747  infmsup  10516  eqglact  16451  odngen  16796  rrgsuppOLD  18135  psrbagsn  18355  coe1mul2lem2  18504  pjdm  18911  xkoccn  20286  txcnmpt  20291  txdis1cn  20302  pthaus  20305  txkgen  20319  xkoco1cn  20324  xkoco2cn  20325  xkoinjcn  20354  txcon  20356  imasnopn  20357  imasncld  20358  imasncls  20359  ptcmplem1  20718  ptcmplem3  20720  ptcmplem4  20721  tmdgsum2  20761  symgtgp  20766  tgpconcompeqg  20776  ghmcnp  20779  tgpt0  20783  qustgpopn  20784  qustgphaus  20787  eltsms  20797  prdsxmslem2  21198  efopn  23207  atansopn  23460  xrlimcnp  23496  suppss2f  27698  fpwrelmapffslem  27786  ptrest  30288  mbfposadd  30302  cnambfre  30303  itg2addnclem2  30307  iblabsnclem  30318  ftc1anclem1  30330  ftc1anclem6  30335  pwfi2f1o  31283  pwfi2f1oOLD  31284