Theorem mptpreima 4757

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 3811	. . . . . 6 |- (x e. A |-> B) = { <.x , y >. | (x e. A /\ y = B) }
3	1, 2	eqtri 2057	. . . . 5 |- F = { <.x , y >. | (x e. A /\ y = B) }
4	3	cnveqi 4453	. . . 4 |- `' F = `' { <.x , y >. | (x e. A /\ y = B) }
5		cnvopab 4669	. . . 4 |- `' { <.x , y >. | (x e. A /\ y = B) } = { <.y , x >. | (x e. A /\ y = B) }
6	4, 5	eqtri 2057	. . 3 |- `' F = { <.y , x >. | (x e. A /\ y = B) }
7	6	imaeq1i 4608	. 2 |- ( `' F " C) = ( { <.y , x >. | (x e. A /\ y = B) } " C)
8		df-ima 4301	. . 3 |- ( { <.y , x >. | (x e. A /\ y = B) } " C) = ran ( { <.y , x >. | (x e. A /\ y = B) } |` C)
9		resopab 4595	. . . . 5 |- ( { <.y , x >. | (x e. A /\ y = B) } |` C) = { <.y , x >. | (y e. C /\ (x e. A /\ y = B)) }
10	9	rneqi 4505	. . . 4 |- ran ( { <.y , x >. | (x e. A /\ y = B) } |` C) = ran { <.y , x >. | (y e. C /\ (x e. A /\ y = B)) }
11		ancom 253	. . . . . . . . 9 |- ((y e. C /\ (x e. A /\ y = B)) <-> ((x e. A /\ y = B) /\ y e. C))
12		anass 381	. . . . . . . . 9 |- (((x e. A /\ y = B) /\ y e. C) <-> (x e. A /\ (y = B /\ y e. C)))
13	11, 12	bitri 173	. . . . . . . 8 |- ((y e. C /\ (x e. A /\ y = B)) <-> (x e. A /\ (y = B /\ y e. C)))
14	13	exbii 1493	. . . . . . 7 |- (E.y(y e. C /\ (x e. A /\ y = B)) <-> E.y(x e. A /\ (y = B /\ y e. C)))
15		19.42v 1783	. . . . . . . 8 |- (E.y(x e. A /\ (y = B /\ y e. C)) <-> (x e. A /\ E.y(y = B /\ y e. C)))
16		df-clel 2033	. . . . . . . . . 10 |- (B e. C <-> E.y(y = B /\ y e. C))
17	16	bicomi 123	. . . . . . . . 9 |- (E.y(y = B /\ y e. C) <-> B e. C)
18	17	anbi2i 430	. . . . . . . 8 |- ((x e. A /\ E.y(y = B /\ y e. C)) <-> (x e. A /\ B e. C))
19	15, 18	bitri 173	. . . . . . 7 |- (E.y(x e. A /\ (y = B /\ y e. C)) <-> (x e. A /\ B e. C))
20	14, 19	bitri 173	. . . . . 6 |- (E.y(y e. C /\ (x e. A /\ y = B)) <-> (x e. A /\ B e. C))
21	20	abbii 2150	. . . . 5 |- {x | E.y(y e. C /\ (x e. A /\ y = B)) } = {x | (x e. A /\ B e. C) }
22		rnopab 4524	. . . . 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 2309	. . . . 5 |- {x e. A | B e. C } = {x | (x e. A /\ B e. C) }
24	21, 22, 23	3eqtr4i 2067	. . . 4 |- ran { <.y , x >. | (y e. C /\ (x e. A /\ y = B)) } = {x e. A | B e. C }
25	10, 24	eqtri 2057	. . 3 |- ran ( { <.y , x >. | (x e. A /\ y = B) } |` C) = {x e. A | B e. C }
26	8, 25	eqtri 2057	. 2 |- ( { <.y , x >. | (x e. A /\ y = B) } " C) = {x e. A | B e. C }
27	7, 26	eqtri 2057	1 |- ( `' F " C) = {x e. A | B e. C }

Syntax hints:   /\ wa 97   = wceq 1242  E.wex 1378   e. wcel 1390   {cab 2023   {crab 2304   {copab 3808   |-> cmpt 3809   `'ccnv 4287   ran crn 4289   |` cres 4290   "cima 4291

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301

This theorem is referenced by:  mptiniseg  4758  dmmpt  4759  fmpt  5262  f1oresrab  5272  suppssfv  5650  suppssov1  5651