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Theorem rnmpt 4525
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 = (x AB)
Assertion
Ref Expression
rnmpt ran 𝐹 = {yx A 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 4524 . 2 ran {⟨x, y⟩ ∣ (x A y = B)} = {yx(x A y = B)}
2 rnmpt.1 . . . 4 𝐹 = (x AB)
3 df-mpt 3811 . . . 4 (x AB) = {⟨x, y⟩ ∣ (x A y = B)}
42, 3eqtri 2057 . . 3 𝐹 = {⟨x, y⟩ ∣ (x A y = B)}
54rneqi 4505 . 2 ran 𝐹 = ran {⟨x, y⟩ ∣ (x A y = B)}
6 df-rex 2306 . . 3 (x A y = Bx(x A y = B))
76abbii 2150 . 2 {yx A y = B} = {yx(x A y = B)}
81, 5, 73eqtr4i 2067 1 ran 𝐹 = {yx A y = B}
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wrex 2301  {copab 3808  cmpt 3809  ran crn 4289
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-cnv 4296  df-dm 4298  df-rn 4299
This theorem is referenced by:  elrnmpt  4526  elrnmpt1  4528  elrnmptg  4529  dfiun3g  4532  dfiin3g  4533  fnrnfv  5163  fmpt  5262  fnasrn  5284  fnasrng  5286  fliftf  5382  abrexex  5686  abrexexg  5687  fo1st  5726  fo2nd  5727  qliftf  6127
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