Theorem mptpreima 5331

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 4352	. . . . . 6 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
3	1, 2	eqtri 2463	. . . . 5 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
4	3	cnveqi 5014	. . . 4 |- `' F = `' { <. x , y >. | ( x e. A /\ y = B ) }
5		cnvopab 5238	. . . 4 |- `' { <. x , y >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( x e. A /\ y = B ) }
6	4, 5	eqtri 2463	. . 3 |- `' F = { <. y , x >. | ( x e. A /\ y = B ) }
7	6	imaeq1i 5166	. 2 |- ( `' F " C ) = ( { <. y , x >. | ( x e. A /\ y = B ) } " C )
8		df-ima 4853	. . 3 |- ( { <. y , x >. | ( x e. A /\ y = B ) } " C ) = ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C )
9		resopab 5153	. . . . 5 |- ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) }
10	9	rneqi 5066	. . . 4 |- ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = ran { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) }
11		ancom 450	. . . . . . . . 9 |- ( ( y e. C /\ ( x e. A /\ y = B ) ) <-> ( ( x e. A /\ y = B ) /\ y e. C ) )
12		anass 649	. . . . . . . . 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 1634	. . . . . . 7 |- ( E. y ( y e. C /\ ( x e. A /\ y = B ) ) <-> E. y ( x e. A /\ ( y = B /\ y e. C ) ) )
15		19.42v 1924	. . . . . . . 8 |- ( E. y ( x e. A /\ ( y = B /\ y e. C ) ) <-> ( x e. A /\ E. y ( y = B /\ y e. C ) ) )
16		df-clel 2439	. . . . . . . . . 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 694	. . . . . . . 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 2555	. . . . 5 |- { x | E. y ( y e. C /\ ( x e. A /\ y = B ) ) } = { x | ( x e. A /\ B e. C ) }
22		rnopab 5084	. . . . 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 2724	. . . . 5 |- { x e. A | B e. C } = { x | ( x e. A /\ B e. C ) }
24	21, 22, 23	3eqtr4i 2473	. . . 4 |- ran { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) } = { x e. A | B e. C }
25	10, 24	eqtri 2463	. . 3 |- ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = { x e. A | B e. C }
26	8, 25	eqtri 2463	. 2 |- ( { <. y , x >. | ( x e. A /\ y = B ) } " C ) = { x e. A | B e. C }
27	7, 26	eqtri 2463	1 |- ( `' F " C ) = { x e. A | B e. C }

Syntax hints:   /\ wa 369   = wceq 1369   E.wex 1586   e. wcel 1756   {cab 2429   {crab 2719   {copab 4349   e. cmpt 4350   `'ccnv 4839   ran crn 4841   |` cres 4842   "cima 4843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-mpt 4352  df-xp 4846  df-rel 4847  df-cnv 4848  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853

This theorem is referenced by:  mptiniseg  5332  dmmpt  5333  fmpt  5864  f1oresrab  5875  suppss2OLD  6315  suppssfvOLD  6316  suppssov1OLD  6317  mptsuppdifd  6711  cantnflem1OLD  7920  cantnfOLD  7923  r0weon  8179  compss  8545  infmsup  10308  eqglact  15732  odngen  16076  rrgsuppOLD  17363  psrbagsn  17577  coe1mul2lem2  17722  pjdm  18132  xkoccn  19192  txcnmpt  19197  txdis1cn  19208  pthaus  19211  txkgen  19225  xkoco1cn  19230  xkoco2cn  19231  xkoinjcn  19260  txcon  19262  imasnopn  19263  imasncld  19264  imasncls  19265  ptcmplem1  19624  ptcmplem3  19626  ptcmplem4  19627  tmdgsum2  19667  symgtgp  19672  tgpconcompeqg  19682  ghmcnp  19685  tgpt0  19689  divstgpopn  19690  divstgphaus  19693  eltsms  19703  prdsxmslem2  20104  efopn  22103  atansopn  22327  xrlimcnp  22362  suppss2f  25954  fpwrelmapffslem  26032  ptrest  28425  mbfposadd  28439  cnambfre  28440  itg2addnclem2  28444  iblabsnclem  28455  ftc1anclem1  28467  ftc1anclem6  28472  pwfi2f1o  29451