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Theorem rnmpo 5931
Description: The range of an operation given by the maps-to notation. (Contributed by FL, 20-Jun-2011.)
Hypothesis
Ref Expression
rngop.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
rnmpo ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
Distinct variable groups:   𝑦,𝑧,𝐴   𝑧,𝐵   𝑧,𝐶   𝑧,𝐹   𝑥,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem rnmpo
StepHypRef Expression
1 rngop.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 df-mpo 5829 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2178 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
43rneqi 4814 . 2 ran 𝐹 = ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
5 rnoprab2 5905 . 2 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
64, 5eqtri 2178 1 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1335  wcel 2128  {cab 2143  wrex 2436  ran crn 4587  {coprab 5825  cmpo 5826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-br 3966  df-opab 4026  df-cnv 4594  df-dm 4596  df-rn 4597  df-oprab 5828  df-mpo 5829
This theorem is referenced by:  elrnmpog  5933  elrnmpo  5934  ralrnmpo  5935  rexrnmpo  5936  txuni2  12656  txbas  12658  txrest  12676
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