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Theorem rnmpt 4630
 Description: The range of a function in maps-to notation. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
rnmpt.1 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
rnmpt ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥,𝑦)

Proof of Theorem rnmpt
StepHypRef Expression
1 rnopab 4629 . 2 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
2 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
3 df-mpt 3861 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
42, 3eqtri 2103 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
54rneqi 4610 . 2 ran 𝐹 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
6 df-rex 2359 . . 3 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
76abbii 2198 . 2 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
81, 5, 73eqtr4i 2113 1 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
 Colors of variables: wff set class Syntax hints:   ∧ wa 102   = wceq 1285  ∃wex 1422   ∈ wcel 1434  {cab 2069  ∃wrex 2354  {copab 3858   ↦ cmpt 3859  ran crn 4392 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-mpt 3861  df-cnv 4399  df-dm 4401  df-rn 4402 This theorem is referenced by:  elrnmpt  4631  elrnmpt1  4633  elrnmptg  4634  dfiun3g  4637  dfiin3g  4638  fnrnfv  5272  fmpt  5371  fnasrn  5393  fnasrng  5395  fliftf  5490  abrexex  5795  abrexexg  5796  fo1st  5835  fo2nd  5836  qliftf  6278  negfi  10311
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