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 ∈ A ↦ B)

Assertion

Ref	Expression
mptpreima	⊢ (◡𝐹 “ 𝐶) = {x ∈ A ∣ B ∈ 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 ∈ A ↦ B)
2		df-mpt 3811	. . . . . 6 ⊢ (x ∈ A ↦ B) = {⟨x, y⟩ ∣ (x ∈ A ∧ y = B)}
3	1, 2	eqtri 2057	. . . . 5 ⊢ 𝐹 = {⟨x, y⟩ ∣ (x ∈ A ∧ y = B)}
4	3	cnveqi 4453	. . . 4 ⊢ ◡𝐹 = ◡{⟨x, y⟩ ∣ (x ∈ A ∧ y = B)}
5		cnvopab 4669	. . . 4 ⊢ ◡{⟨x, y⟩ ∣ (x ∈ A ∧ y = B)} = {⟨y, x⟩ ∣ (x ∈ A ∧ y = B)}
6	4, 5	eqtri 2057	. . 3 ⊢ ◡𝐹 = {⟨y, x⟩ ∣ (x ∈ A ∧ y = B)}
7	6	imaeq1i 4608	. 2 ⊢ (◡𝐹 “ 𝐶) = ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} “ 𝐶)
8		df-ima 4301	. . 3 ⊢ ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} “ 𝐶) = ran ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} ↾ 𝐶)
9		resopab 4595	. . . . 5 ⊢ ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} ↾ 𝐶) = {⟨y, x⟩ ∣ (y ∈ 𝐶 ∧ (x ∈ A ∧ y = B))}
10	9	rneqi 4505	. . . 4 ⊢ ran ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} ↾ 𝐶) = ran {⟨y, x⟩ ∣ (y ∈ 𝐶 ∧ (x ∈ A ∧ y = B))}
11		ancom 253	. . . . . . . . 9 ⊢ ((y ∈ 𝐶 ∧ (x ∈ A ∧ y = B)) ↔ ((x ∈ A ∧ y = B) ∧ y ∈ 𝐶))
12		anass 381	. . . . . . . . 9 ⊢ (((x ∈ A ∧ y = B) ∧ y ∈ 𝐶) ↔ (x ∈ A ∧ (y = B ∧ y ∈ 𝐶)))
13	11, 12	bitri 173	. . . . . . . 8 ⊢ ((y ∈ 𝐶 ∧ (x ∈ A ∧ y = B)) ↔ (x ∈ A ∧ (y = B ∧ y ∈ 𝐶)))
14	13	exbii 1493	. . . . . . 7 ⊢ (∃y(y ∈ 𝐶 ∧ (x ∈ A ∧ y = B)) ↔ ∃y(x ∈ A ∧ (y = B ∧ y ∈ 𝐶)))
15		19.42v 1783	. . . . . . . 8 ⊢ (∃y(x ∈ A ∧ (y = B ∧ y ∈ 𝐶)) ↔ (x ∈ A ∧ ∃y(y = B ∧ y ∈ 𝐶)))
16		df-clel 2033	. . . . . . . . . 10 ⊢ (B ∈ 𝐶 ↔ ∃y(y = B ∧ y ∈ 𝐶))
17	16	bicomi 123	. . . . . . . . 9 ⊢ (∃y(y = B ∧ y ∈ 𝐶) ↔ B ∈ 𝐶)
18	17	anbi2i 430	. . . . . . . 8 ⊢ ((x ∈ A ∧ ∃y(y = B ∧ y ∈ 𝐶)) ↔ (x ∈ A ∧ B ∈ 𝐶))
19	15, 18	bitri 173	. . . . . . 7 ⊢ (∃y(x ∈ A ∧ (y = B ∧ y ∈ 𝐶)) ↔ (x ∈ A ∧ B ∈ 𝐶))
20	14, 19	bitri 173	. . . . . 6 ⊢ (∃y(y ∈ 𝐶 ∧ (x ∈ A ∧ y = B)) ↔ (x ∈ A ∧ B ∈ 𝐶))
21	20	abbii 2150	. . . . 5 ⊢ {x ∣ ∃y(y ∈ 𝐶 ∧ (x ∈ A ∧ y = B))} = {x ∣ (x ∈ A ∧ B ∈ 𝐶)}
22		rnopab 4524	. . . . 5 ⊢ ran {⟨y, x⟩ ∣ (y ∈ 𝐶 ∧ (x ∈ A ∧ y = B))} = {x ∣ ∃y(y ∈ 𝐶 ∧ (x ∈ A ∧ y = B))}
23		df-rab 2309	. . . . 5 ⊢ {x ∈ A ∣ B ∈ 𝐶} = {x ∣ (x ∈ A ∧ B ∈ 𝐶)}
24	21, 22, 23	3eqtr4i 2067	. . . 4 ⊢ ran {⟨y, x⟩ ∣ (y ∈ 𝐶 ∧ (x ∈ A ∧ y = B))} = {x ∈ A ∣ B ∈ 𝐶}
25	10, 24	eqtri 2057	. . 3 ⊢ ran ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} ↾ 𝐶) = {x ∈ A ∣ B ∈ 𝐶}
26	8, 25	eqtri 2057	. 2 ⊢ ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} “ 𝐶) = {x ∈ A ∣ B ∈ 𝐶}
27	7, 26	eqtri 2057	1 ⊢ (◡𝐹 “ 𝐶) = {x ∈ A ∣ B ∈ 𝐶}

Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ 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-bnd 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