Theorem mptpreima 6191

Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)

Hypothesis

Ref	Expression
dmmpt.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		dmmpt.1	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
2		df-mpt 5177	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
3	1, 2	eqtri 2752	. . . . 5 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4	3	cnveqi 5821	. . . 4 ⊢ ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
5		cnvopab 6090	. . . 4 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	4, 5	eqtri 2752	. . 3 ⊢ ◡𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
7	6	imaeq1i 6012	. 2 ⊢ (◡𝐹 “ 𝐶) = ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶)
8		df-ima 5636	. . 3 ⊢ ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) = ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶)
9		resopab 5989	. . . . 5 ⊢ ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))}
10	9	rneqi 5883	. . . 4 ⊢ ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))}
11		ancom 460	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑦 ∈ 𝐶))
12		anass 468	. . . . . . . . 9 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ∧ 𝑦 ∈ 𝐶) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)))
13	11, 12	bitri 275	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)))
14	13	exbii 1848	. . . . . . 7 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)))
15		19.42v 1953	. . . . . . . 8 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)))
16		dfclel 2804	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐶 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶))
17	16	bicomi 224	. . . . . . . . 9 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶) ↔ 𝐵 ∈ 𝐶)
18	17	anbi2i 623	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶))
19	15, 18	bitri 275	. . . . . . 7 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶))
20	14, 19	bitri 275	. . . . . 6 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶))
21	20	abbii 2796	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)}
22		rnopab 5900	. . . . 5 ⊢ ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))}
23		df-rab 3397	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)}
24	21, 22, 23	3eqtr4i 2762	. . . 4 ⊢ ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
25	10, 24	eqtri 2752	. . 3 ⊢ ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
26	8, 25	eqtri 2752	. 2 ⊢ ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
27	7, 26	eqtri 2752	1 ⊢ (◡𝐹 “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2707  {crab 3396  {copab 5157   ↦ cmpt 5176  ^◡ccnv 5622  ran crn 5624   ↾ cres 5625   “ cima 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-mpt 5177  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636

This theorem is referenced by:  mptiniseg  6192  dmmpt  6193  fmpt  7048  f1oresrab  7065  mptsuppdifd  8126  r0weon  9925  compss  10289  infrenegsup  12126  eqglact  19076  odngen  19474  pjdm  21632  psrbagsn  21986  coe1mul2lem2  22170  xkoccn  23522  txcnmpt  23527  txdis1cn  23538  pthaus  23541  txkgen  23555  xkoco1cn  23560  xkoco2cn  23561  xkoinjcn  23590  txconn  23592  imasnopn  23593  imasncld  23594  imasncls  23595  ptcmplem1  23955  ptcmplem3  23957  ptcmplem4  23958  tmdgsum2  23999  symgtgp  24009  tgpconncompeqg  24015  ghmcnp  24018  tgpt0  24022  qustgpopn  24023  qustgphaus  24026  eltsms  24036  prdsxmslem2  24433  efopn  26583  atansopn  26858  xrlimcnp  26894  fpwrelmapffslem  32688  ptrest  37598  mbfposadd  37646  cnambfre  37647  itg2addnclem2  37651  iblabsnclem  37662  ftc1anclem1  37672  ftc1anclem6  37677  resuppsinopn  42336  pwfi2f1o  43069  smfpimioo  46769