Theorem mptpreima 5683

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 5653	. . . . . 6 ⊢ (x ∈ A ↦ B) = {⟨x, y⟩ ∣ (x ∈ A ∧ y = B)}
3	1, 2	eqtri 2373	. . . . 5 ⊢ F = {⟨x, y⟩ ∣ (x ∈ A ∧ y = B)}
4	3	cnveqi 4888	. . . 4 ⊢ ◡F = ◡{⟨x, y⟩ ∣ (x ∈ A ∧ y = B)}
5		cnvopab 5031	. . . 4 ⊢ ◡{⟨x, y⟩ ∣ (x ∈ A ∧ y = B)} = {⟨y, x⟩ ∣ (x ∈ A ∧ y = B)}
6	4, 5	eqtri 2373	. . 3 ⊢ ◡F = {⟨y, x⟩ ∣ (x ∈ A ∧ y = B)}
7	6	imaeq1i 4940	. 2 ⊢ (◡F “ C) = ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} “ C)
8		dfima3 4952	. . 3 ⊢ ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} “ C) = ran ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} ↾ C)
9		resopab 5000	. . . . 5 ⊢ ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} ↾ C) = {⟨y, x⟩ ∣ (y ∈ C ∧ (x ∈ A ∧ y = B))}
10	9	rneqi 4958	. . . 4 ⊢ ran ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} ↾ C) = ran {⟨y, x⟩ ∣ (y ∈ C ∧ (x ∈ A ∧ y = B))}
11		ancom 437	. . . . . . . . 9 ⊢ ((y ∈ C ∧ (x ∈ A ∧ y = B)) ↔ ((x ∈ A ∧ y = B) ∧ y ∈ C))
12		anass 630	. . . . . . . . 9 ⊢ (((x ∈ A ∧ y = B) ∧ y ∈ C) ↔ (x ∈ A ∧ (y = B ∧ y ∈ C)))
13	11, 12	bitri 240	. . . . . . . 8 ⊢ ((y ∈ C ∧ (x ∈ A ∧ y = B)) ↔ (x ∈ A ∧ (y = B ∧ y ∈ C)))
14	13	exbii 1582	. . . . . . 7 ⊢ (∃y(y ∈ C ∧ (x ∈ A ∧ y = B)) ↔ ∃y(x ∈ A ∧ (y = B ∧ y ∈ C)))
15		19.42v 1905	. . . . . . . 8 ⊢ (∃y(x ∈ A ∧ (y = B ∧ y ∈ C)) ↔ (x ∈ A ∧ ∃y(y = B ∧ y ∈ C)))
16		df-clel 2349	. . . . . . . . . 10 ⊢ (B ∈ C ↔ ∃y(y = B ∧ y ∈ C))
17	16	bicomi 193	. . . . . . . . 9 ⊢ (∃y(y = B ∧ y ∈ C) ↔ B ∈ C)
18	17	anbi2i 675	. . . . . . . 8 ⊢ ((x ∈ A ∧ ∃y(y = B ∧ y ∈ C)) ↔ (x ∈ A ∧ B ∈ C))
19	15, 18	bitri 240	. . . . . . 7 ⊢ (∃y(x ∈ A ∧ (y = B ∧ y ∈ C)) ↔ (x ∈ A ∧ B ∈ C))
20	14, 19	bitri 240	. . . . . 6 ⊢ (∃y(y ∈ C ∧ (x ∈ A ∧ y = B)) ↔ (x ∈ A ∧ B ∈ C))
21	20	abbii 2466	. . . . 5 ⊢ {x ∣ ∃y(y ∈ C ∧ (x ∈ A ∧ y = B))} = {x ∣ (x ∈ A ∧ B ∈ C)}
22		rnopab 4968	. . . . 5 ⊢ ran {⟨y, x⟩ ∣ (y ∈ C ∧ (x ∈ A ∧ y = B))} = {x ∣ ∃y(y ∈ C ∧ (x ∈ A ∧ y = B))}
23		df-rab 2624	. . . . 5 ⊢ {x ∈ A ∣ B ∈ C} = {x ∣ (x ∈ A ∧ B ∈ C)}
24	21, 22, 23	3eqtr4i 2383	. . . 4 ⊢ ran {⟨y, x⟩ ∣ (y ∈ C ∧ (x ∈ A ∧ y = B))} = {x ∈ A ∣ B ∈ C}
25	10, 24	eqtri 2373	. . 3 ⊢ ran ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} ↾ C) = {x ∈ A ∣ B ∈ C}
26	8, 25	eqtri 2373	. 2 ⊢ ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} “ C) = {x ∈ A ∣ B ∈ C}
27	7, 26	eqtri 2373	1 ⊢ (◡F “ C) = {x ∈ A ∣ B ∈ C}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2619  {copab 4623   “ cima 4723  ^◡ccnv 4772  ran crn 4774   ↾ cres 4775   ↦ cmpt 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-res 4789  df-mpt 5653

This theorem is referenced by:  fmpt  5693