Theorem mptpreima 6238

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 5206	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
3	1, 2	eqtri 2757	. . . . 5 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
4	3	cnveqi 5865	. . . 4 ⊢ ◡𝐹 = ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
5		cnvopab 6137	. . . 4 ⊢ ◡{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
6	4, 5	eqtri 2757	. . 3 ⊢ ◡𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
7	6	imaeq1i 6055	. 2 ⊢ (◡𝐹 “ 𝐶) = ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶)
8		df-ima 5678	. . 3 ⊢ ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) = ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶)
9		resopab 6032	. . . . 5 ⊢ ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))}
10	9	rneqi 5928	. . . 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 1847	. . . . . . 7 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) ↔ ∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)))
15		19.42v 1952	. . . . . . . 8 ⊢ (∃𝑦(𝑥 ∈ 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ (𝑥 ∈ 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝑦 ∈ 𝐶)))
16		dfclel 2809	. . . . . . . . . 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 2801	. . . . 5 ⊢ {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)}
22		rnopab 5945	. . . . 5 ⊢ ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))}
23		df-rab 3420	. . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶)}
24	21, 22, 23	3eqtr4i 2767	. . . 4 ⊢ ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵))} = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
25	10, 24	eqtri 2757	. . 3 ⊢ ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↾ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
26	8, 25	eqtri 2757	. 2 ⊢ ({⟨𝑦, 𝑥⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}
27	7, 26	eqtri 2757	1 ⊢ (◡𝐹 “ 𝐶) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ 𝐶}

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2712  {crab 3419  {copab 5185   ↦ cmpt 5205  ^◡ccnv 5664  ran crn 5666   ↾ cres 5667   “ cima 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5124  df-opab 5186  df-mpt 5206  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678

This theorem is referenced by:  mptiniseg  6239  dmmpt  6240  fmpt  7110  f1oresrab  7127  mptsuppdifd  8193  r0weon  10034  compss  10398  infrenegsup  12233  eqglact  19167  odngen  19564  pjdm  21682  psrbagsn  22036  coe1mul2lem2  22220  xkoccn  23574  txcnmpt  23579  txdis1cn  23590  pthaus  23593  txkgen  23607  xkoco1cn  23612  xkoco2cn  23613  xkoinjcn  23642  txconn  23644  imasnopn  23645  imasncld  23646  imasncls  23647  ptcmplem1  24007  ptcmplem3  24009  ptcmplem4  24010  tmdgsum2  24051  symgtgp  24061  tgpconncompeqg  24067  ghmcnp  24070  tgpt0  24074  qustgpopn  24075  qustgphaus  24078  eltsms  24088  prdsxmslem2  24487  efopn  26637  atansopn  26912  xrlimcnp  26948  fpwrelmapffslem  32679  ptrest  37601  mbfposadd  37649  cnambfre  37650  itg2addnclem2  37654  iblabsnclem  37665  ftc1anclem1  37675  ftc1anclem6  37680  resuppsinopn  42372  pwfi2f1o  43086  smfpimioo  46774