Theorem mptpreima 5682

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 5652	. . . . . 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 4887	. . . 4 ⊢ ◡F = ◡{⟨x, y⟩ ∣ (x ∈ A ∧ y = B)}
5		cnvopab 5030	. . . 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 4939	. 2 ⊢ (◡F “ C) = ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} “ C)
8		dfima3 4951	. . 3 ⊢ ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} “ C) = ran ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} ↾ C)
9		resopab 4999	. . . . 5 ⊢ ({⟨y, x⟩ ∣ (x ∈ A ∧ y = B)} ↾ C) = {⟨y, x⟩ ∣ (y ∈ C ∧ (x ∈ A ∧ y = B))}
10	9	rneqi 4957	. . . 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 2465	. . . . 5 ⊢ {x ∣ ∃y(y ∈ C ∧ (x ∈ A ∧ y = B))} = {x ∣ (x ∈ A ∧ B ∈ C)}
22		rnopab 4967	. . . . 5 ⊢ ran {⟨y, x⟩ ∣ (y ∈ C ∧ (x ∈ A ∧ y = B))} = {x ∣ ∃y(y ∈ C ∧ (x ∈ A ∧ y = B))}
23		df-rab 2623	. . . . 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 2618  {copab 4622   “ cima 4722  ^◡ccnv 4771  ran crn 4773   ↾ cres 4774   ↦ cmpt 5651

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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-res 4788  df-mpt 5652

This theorem is referenced by:  fmpt  5692