Theorem mptpreima 5500

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 4507	. . . . . 6 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
3	1, 2	eqtri 2496	. . . . 5 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
4	3	cnveqi 5177	. . . 4 |- `' F = `' { <. x , y >. | ( x e. A /\ y = B ) }
5		cnvopab 5407	. . . 4 |- `' { <. x , y >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( x e. A /\ y = B ) }
6	4, 5	eqtri 2496	. . 3 |- `' F = { <. y , x >. | ( x e. A /\ y = B ) }
7	6	imaeq1i 5334	. 2 |- ( `' F " C ) = ( { <. y , x >. | ( x e. A /\ y = B ) } " C )
8		df-ima 5012	. . 3 |- ( { <. y , x >. | ( x e. A /\ y = B ) } " C ) = ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C )
9		resopab 5320	. . . . 5 |- ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) }
10	9	rneqi 5229	. . . 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 1644	. . . . . . 7 |- ( E. y ( y e. C /\ ( x e. A /\ y = B ) ) <-> E. y ( x e. A /\ ( y = B /\ y e. C ) ) )
15		19.42v 1949	. . . . . . . 8 |- ( E. y ( x e. A /\ ( y = B /\ y e. C ) ) <-> ( x e. A /\ E. y ( y = B /\ y e. C ) ) )
16		df-clel 2462	. . . . . . . . . 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 2601	. . . . 5 |- { x | E. y ( y e. C /\ ( x e. A /\ y = B ) ) } = { x | ( x e. A /\ B e. C ) }
22		rnopab 5247	. . . . 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 2823	. . . . 5 |- { x e. A | B e. C } = { x | ( x e. A /\ B e. C ) }
24	21, 22, 23	3eqtr4i 2506	. . . 4 |- ran { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) } = { x e. A | B e. C }
25	10, 24	eqtri 2496	. . 3 |- ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = { x e. A | B e. C }
26	8, 25	eqtri 2496	. 2 |- ( { <. y , x >. | ( x e. A /\ y = B ) } " C ) = { x e. A | B e. C }
27	7, 26	eqtri 2496	1 |- ( `' F " C ) = { x e. A | B e. C }

Syntax hints:   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   {cab 2452   {crab 2818   {copab 4504   |-> cmpt 4505   `'ccnv 4998   ran crn 5000   |` cres 5001   "cima 5002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012

This theorem is referenced by:  mptiniseg  5501  dmmpt  5502  fmpt  6042  f1oresrab  6053  suppss2OLD  6514  suppssfvOLD  6515  suppssov1OLD  6516  mptsuppdifd  6922  cantnflem1OLD  8131  cantnfOLD  8134  r0weon  8390  compss  8756  infmsup  10521  eqglact  16057  odngen  16403  rrgsuppOLD  17739  psrbagsn  17959  coe1mul2lem2  18108  pjdm  18533  xkoccn  19883  txcnmpt  19888  txdis1cn  19899  pthaus  19902  txkgen  19916  xkoco1cn  19921  xkoco2cn  19922  xkoinjcn  19951  txcon  19953  imasnopn  19954  imasncld  19955  imasncls  19956  ptcmplem1  20315  ptcmplem3  20317  ptcmplem4  20318  tmdgsum2  20358  symgtgp  20363  tgpconcompeqg  20373  ghmcnp  20376  tgpt0  20380  divstgpopn  20381  divstgphaus  20384  eltsms  20394  prdsxmslem2  20795  efopn  22795  atansopn  23019  xrlimcnp  23054  suppss2f  27178  fpwrelmapffslem  27255  ptrest  29653  mbfposadd  29667  cnambfre  29668  itg2addnclem2  29672  iblabsnclem  29683  ftc1anclem1  29695  ftc1anclem6  29700  pwfi2f1o  30676