ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rnmpt GIF version

Theorem rnmpt 4609
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 4608 . 2 ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
2 rnmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐵)
3 df-mpt 3847 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
42, 3eqtri 2076 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
54rneqi 4589 . 2 ran 𝐹 = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
6 df-rex 2329 . . 3 (∃𝑥𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥𝐴𝑦 = 𝐵))
76abbii 2169 . 2 {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥𝐴𝑦 = 𝐵)}
81, 5, 73eqtr4i 2086 1 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵}
Colors of variables: wff set class
Syntax hints:  wa 101   = wceq 1259  wex 1397  wcel 1409  {cab 2042  wrex 2324  {copab 3844  cmpt 3845  ran crn 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-mpt 3847  df-cnv 4380  df-dm 4382  df-rn 4383
This theorem is referenced by:  elrnmpt  4610  elrnmpt1  4612  elrnmptg  4613  dfiun3g  4616  dfiin3g  4617  fnrnfv  5247  fmpt  5346  fnasrn  5368  fnasrng  5370  fliftf  5466  abrexex  5771  abrexexg  5772  fo1st  5811  fo2nd  5812  qliftf  6221
  Copyright terms: Public domain W3C validator