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Theorem mptpreima 5869
Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypothesis
Ref Expression
dmmpt.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
mptpreima (𝐹𝐶) = {𝑥𝐴𝐵𝐶}
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.
StepHypRef Expression
1 dmmpt.1 . . . . . 6 𝐹 = (𝑥𝐴𝐵)
2 df-mpt 4953 . . . . . 6 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
31, 2eqtri 2849 . . . . 5 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
43cnveqi 5529 . . . 4 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
5 cnvopab 5775 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
64, 5eqtri 2849 . . 3 𝐹 = {⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
76imaeq1i 5704 . 2 (𝐹𝐶) = ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶)
8 df-ima 5355 . . 3 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶) = ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶)
9 resopab 5683 . . . . 5 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
109rneqi 5584 . . . 4 ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
11 ancom 454 . . . . . . . . 9 ((𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ ((𝑥𝐴𝑦 = 𝐵) ∧ 𝑦𝐶))
12 anass 462 . . . . . . . . 9 (((𝑥𝐴𝑦 = 𝐵) ∧ 𝑦𝐶) ↔ (𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
1311, 12bitri 267 . . . . . . . 8 ((𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ (𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
1413exbii 1947 . . . . . . 7 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ ∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)))
15 19.42v 2052 . . . . . . . 8 (∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝑦𝐶)))
16 df-clel 2821 . . . . . . . . . 10 (𝐵𝐶 ↔ ∃𝑦(𝑦 = 𝐵𝑦𝐶))
1716bicomi 216 . . . . . . . . 9 (∃𝑦(𝑦 = 𝐵𝑦𝐶) ↔ 𝐵𝐶)
1817anbi2i 616 . . . . . . . 8 ((𝑥𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴𝐵𝐶))
1915, 18bitri 267 . . . . . . 7 (∃𝑦(𝑥𝐴 ∧ (𝑦 = 𝐵𝑦𝐶)) ↔ (𝑥𝐴𝐵𝐶))
2014, 19bitri 267 . . . . . 6 (∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵)) ↔ (𝑥𝐴𝐵𝐶))
2120abbii 2944 . . . . 5 {𝑥 ∣ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥 ∣ (𝑥𝐴𝐵𝐶)}
22 rnopab 5603 . . . . 5 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥 ∣ ∃𝑦(𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))}
23 df-rab 3126 . . . . 5 {𝑥𝐴𝐵𝐶} = {𝑥 ∣ (𝑥𝐴𝐵𝐶)}
2421, 22, 233eqtr4i 2859 . . . 4 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐶 ∧ (𝑥𝐴𝑦 = 𝐵))} = {𝑥𝐴𝐵𝐶}
2510, 24eqtri 2849 . . 3 ran ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↾ 𝐶) = {𝑥𝐴𝐵𝐶}
268, 25eqtri 2849 . 2 ({⟨𝑦, 𝑥⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} “ 𝐶) = {𝑥𝐴𝐵𝐶}
277, 26eqtri 2849 1 (𝐹𝐶) = {𝑥𝐴𝐵𝐶}
Colors of variables: wff setvar class
Syntax hints:  wa 386   = wceq 1656  wex 1878  wcel 2164  {cab 2811  {crab 3121  {copab 4935  cmpt 4952  ccnv 5341  ran crn 5343  cres 5344  cima 5345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-mpt 4953  df-xp 5348  df-rel 5349  df-cnv 5350  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355
This theorem is referenced by:  mptiniseg  5870  dmmpt  5871  fmpt  6629  f1oresrab  6644  mptsuppdifd  7581  r0weon  9148  compss  9513  infrenegsup  11336  eqglact  17996  odngen  18343  psrbagsn  19855  coe1mul2lem2  19998  pjdm  20414  xkoccn  21793  txcnmpt  21798  txdis1cn  21809  pthaus  21812  txkgen  21826  xkoco1cn  21831  xkoco2cn  21832  xkoinjcn  21861  txconn  21863  imasnopn  21864  imasncld  21865  imasncls  21866  ptcmplem1  22226  ptcmplem3  22228  ptcmplem4  22229  tmdgsum2  22270  symgtgp  22275  tgpconncompeqg  22285  ghmcnp  22288  tgpt0  22292  qustgpopn  22293  qustgphaus  22296  eltsms  22306  prdsxmslem2  22704  efopn  24803  atansopn  25072  xrlimcnp  25108  fpwrelmapffslem  30044  ptrest  33945  mbfposadd  33993  cnambfre  33994  itg2addnclem2  33998  iblabsnclem  34009  ftc1anclem1  34021  ftc1anclem6  34026  pwfi2f1o  38502  smfpimioo  41781
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