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Theorem rnmpt2 7002
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
rnmpt2 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 rnmpt2
StepHypRef Expression
1 rngop.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
2 df-mpt2 6881 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
31, 2eqtri 2819 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
43rneqi 5553 . 2 ran 𝐹 = ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
5 rnoprab2 6976 . 2 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
64, 5eqtri 2819 1 ran 𝐹 = {𝑧 ∣ ∃𝑥𝐴𝑦𝐵 𝑧 = 𝐶}
Colors of variables: wff setvar class
Syntax hints:  wa 385   = wceq 1653  wcel 2157  {cab 2783  wrex 3088  ran crn 5311  {coprab 6877  cmpt2 6878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-cnv 5318  df-dm 5320  df-rn 5321  df-oprab 6880  df-mpt2 6881
This theorem is referenced by:  elrnmpt2g  7004  elrnmpt2  7005  ralrnmpt2  7007  dffi3  8577  ixpiunwdom  8736  qnnen  15274  txuni2  21693  txbas  21695  xkobval  21714  xkoopn  21717  txrest  21759  ptrescn  21767  tx1stc  21778  xkoptsub  21782  xkopt  21783  xkococn  21788  ptcmplem4  22183  met2ndci  22651  i1fadd  23799  i1fmul  23800  rnmpt2ss  29982  cnre2csqima  30464  qqhval2  30533  scutf  32423  icoreresf  33689  ptrest  33888  eldiophb  38093
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