Theorem mptpreima 5230

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

Hypothesis

Ref	Expression
dmmpo.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		dmmpo.1	. . . . . 6 |- F = ( x e. A |-> B )
2		df-mpt 4152	. . . . . 6 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
3	1, 2	eqtri 2252	. . . . 5 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
4	3	cnveqi 4905	. . . 4 |- `' F = `' { <. x , y >. | ( x e. A /\ y = B ) }
5		cnvopab 5138	. . . 4 |- `' { <. x , y >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( x e. A /\ y = B ) }
6	4, 5	eqtri 2252	. . 3 |- `' F = { <. y , x >. | ( x e. A /\ y = B ) }
7	6	imaeq1i 5073	. 2 |- ( `' F " C ) = ( { <. y , x >. | ( x e. A /\ y = B ) } " C )
8		df-ima 4738	. . 3 |- ( { <. y , x >. | ( x e. A /\ y = B ) } " C ) = ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C )
9		resopab 5057	. . . . 5 |- ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) }
10	9	rneqi 4960	. . . 4 |- ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = ran { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) }
11		ancom 266	. . . . . . . . 9 |- ( ( y e. C /\ ( x e. A /\ y = B ) ) <-> ( ( x e. A /\ y = B ) /\ y e. C ) )
12		anass 401	. . . . . . . . 9 |- ( ( ( x e. A /\ y = B ) /\ y e. C ) <-> ( x e. A /\ ( y = B /\ y e. C ) ) )
13	11, 12	bitri 184	. . . . . . . 8 |- ( ( y e. C /\ ( x e. A /\ y = B ) ) <-> ( x e. A /\ ( y = B /\ y e. C ) ) )
14	13	exbii 1653	. . . . . . 7 |- ( E. y ( y e. C /\ ( x e. A /\ y = B ) ) <-> E. y ( x e. A /\ ( y = B /\ y e. C ) ) )
15		19.42v 1955	. . . . . . . 8 |- ( E. y ( x e. A /\ ( y = B /\ y e. C ) ) <-> ( x e. A /\ E. y ( y = B /\ y e. C ) ) )
16		df-clel 2227	. . . . . . . . . 10 |- ( B e. C <-> E. y ( y = B /\ y e. C ) )
17	16	bicomi 132	. . . . . . . . 9 |- ( E. y ( y = B /\ y e. C ) <-> B e. C )
18	17	anbi2i 457	. . . . . . . 8 |- ( ( x e. A /\ E. y ( y = B /\ y e. C ) ) <-> ( x e. A /\ B e. C ) )
19	15, 18	bitri 184	. . . . . . 7 |- ( E. y ( x e. A /\ ( y = B /\ y e. C ) ) <-> ( x e. A /\ B e. C ) )
20	14, 19	bitri 184	. . . . . 6 |- ( E. y ( y e. C /\ ( x e. A /\ y = B ) ) <-> ( x e. A /\ B e. C ) )
21	20	abbii 2347	. . . . 5 |- { x | E. y ( y e. C /\ ( x e. A /\ y = B ) ) } = { x | ( x e. A /\ B e. C ) }
22		rnopab 4979	. . . . 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 2519	. . . . 5 |- { x e. A | B e. C } = { x | ( x e. A /\ B e. C ) }
24	21, 22, 23	3eqtr4i 2262	. . . 4 |- ran { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) } = { x e. A | B e. C }
25	10, 24	eqtri 2252	. . 3 |- ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = { x e. A | B e. C }
26	8, 25	eqtri 2252	. 2 |- ( { <. y , x >. | ( x e. A /\ y = B ) } " C ) = { x e. A | B e. C }
27	7, 26	eqtri 2252	1 |- ( `' F " C ) = { x e. A | B e. C }

Syntax hints:   /\ wa 104   = wceq 1397   E.wex 1540   e. wcel 2202   {cab 2217   {crab 2514   {copab 4149   |-> cmpt 4150   `'ccnv 4724   ran crn 4726   |` cres 4727   "cima 4728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  mptiniseg  5231  dmmpt  5232  fmpt  5797  f1oresrab  5812  suppssfv  6230  suppssov1  6231  infrenegsupex  9827  infxrnegsupex  11823  eqglact  13811  fczpsrbag  14684  txcnmpt  14996  txdis1cn  15001  imasnopn  15022