Theorem mptpreima 4834

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 3841	. . . . . 6 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
3	1, 2	eqtri 2101	. . . . 5 |- F = { <. x , y >. | ( x e. A /\ y = B ) }
4	3	cnveqi 4528	. . . 4 |- `' F = `' { <. x , y >. | ( x e. A /\ y = B ) }
5		cnvopab 4746	. . . 4 |- `' { <. x , y >. | ( x e. A /\ y = B ) } = { <. y , x >. | ( x e. A /\ y = B ) }
6	4, 5	eqtri 2101	. . 3 |- `' F = { <. y , x >. | ( x e. A /\ y = B ) }
7	6	imaeq1i 4685	. 2 |- ( `' F " C ) = ( { <. y , x >. | ( x e. A /\ y = B ) } " C )
8		df-ima 4376	. . 3 |- ( { <. y , x >. | ( x e. A /\ y = B ) } " C ) = ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C )
9		resopab 4672	. . . . 5 |- ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) }
10	9	rneqi 4580	. . . 4 |- ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = ran { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) }
11		ancom 262	. . . . . . . . 9 |- ( ( y e. C /\ ( x e. A /\ y = B ) ) <-> ( ( x e. A /\ y = B ) /\ y e. C ) )
12		anass 393	. . . . . . . . 9 |- ( ( ( x e. A /\ y = B ) /\ y e. C ) <-> ( x e. A /\ ( y = B /\ y e. C ) ) )
13	11, 12	bitri 182	. . . . . . . 8 |- ( ( y e. C /\ ( x e. A /\ y = B ) ) <-> ( x e. A /\ ( y = B /\ y e. C ) ) )
14	13	exbii 1536	. . . . . . 7 |- ( E. y ( y e. C /\ ( x e. A /\ y = B ) ) <-> E. y ( x e. A /\ ( y = B /\ y e. C ) ) )
15		19.42v 1827	. . . . . . . 8 |- ( E. y ( x e. A /\ ( y = B /\ y e. C ) ) <-> ( x e. A /\ E. y ( y = B /\ y e. C ) ) )
16		df-clel 2077	. . . . . . . . . 10 |- ( B e. C <-> E. y ( y = B /\ y e. C ) )
17	16	bicomi 130	. . . . . . . . 9 |- ( E. y ( y = B /\ y e. C ) <-> B e. C )
18	17	anbi2i 444	. . . . . . . 8 |- ( ( x e. A /\ E. y ( y = B /\ y e. C ) ) <-> ( x e. A /\ B e. C ) )
19	15, 18	bitri 182	. . . . . . 7 |- ( E. y ( x e. A /\ ( y = B /\ y e. C ) ) <-> ( x e. A /\ B e. C ) )
20	14, 19	bitri 182	. . . . . 6 |- ( E. y ( y e. C /\ ( x e. A /\ y = B ) ) <-> ( x e. A /\ B e. C ) )
21	20	abbii 2194	. . . . 5 |- { x | E. y ( y e. C /\ ( x e. A /\ y = B ) ) } = { x | ( x e. A /\ B e. C ) }
22		rnopab 4599	. . . . 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 2357	. . . . 5 |- { x e. A | B e. C } = { x | ( x e. A /\ B e. C ) }
24	21, 22, 23	3eqtr4i 2111	. . . 4 |- ran { <. y , x >. | ( y e. C /\ ( x e. A /\ y = B ) ) } = { x e. A | B e. C }
25	10, 24	eqtri 2101	. . 3 |- ran ( { <. y , x >. | ( x e. A /\ y = B ) } |` C ) = { x e. A | B e. C }
26	8, 25	eqtri 2101	. 2 |- ( { <. y , x >. | ( x e. A /\ y = B ) } " C ) = { x e. A | B e. C }
27	7, 26	eqtri 2101	1 |- ( `' F " C ) = { x e. A | B e. C }

Syntax hints:   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   {cab 2067   {crab 2352   {copab 3838   |-> cmpt 3839   `'ccnv 4362   ran crn 4364   |` cres 4365   "cima 4366

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-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-mpt 3841  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by:  mptiniseg  4835  dmmpt  4836  fmpt  5340  f1oresrab  5350  suppssfv  5728  suppssov1  5729  infrenegsupex  8682