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Theorem rnopab 4504
Description: The range of a class of ordered pairs. (Contributed by NM, 14-Aug-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
rnopab ran {⟨x, y⟩ ∣ φ} = {yxφ}
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)

Proof of Theorem rnopab
StepHypRef Expression
1 nfopab1 3796 . . 3 x{⟨x, y⟩ ∣ φ}
2 nfopab2 3797 . . 3 y{⟨x, y⟩ ∣ φ}
31, 2dfrnf 4498 . 2 ran {⟨x, y⟩ ∣ φ} = {yx x{⟨x, y⟩ ∣ φ}y}
4 df-br 3735 . . . . 5 (x{⟨x, y⟩ ∣ φ}y ↔ ⟨x, y {⟨x, y⟩ ∣ φ})
5 opabid 3964 . . . . 5 (⟨x, y {⟨x, y⟩ ∣ φ} ↔ φ)
64, 5bitri 173 . . . 4 (x{⟨x, y⟩ ∣ φ}yφ)
76exbii 1474 . . 3 (x x{⟨x, y⟩ ∣ φ}yxφ)
87abbii 2131 . 2 {yx x{⟨x, y⟩ ∣ φ}y} = {yxφ}
93, 8eqtri 2038 1 ran {⟨x, y⟩ ∣ φ} = {yxφ}
Colors of variables: wff set class
Syntax hints:   = wceq 1226  wex 1358   wcel 1370  {cab 2004  cop 3349   class class class wbr 3734  {copab 3787  ran crn 4269
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-cnv 4276  df-dm 4278  df-rn 4279
This theorem is referenced by:  rnmpt  4505  mptpreima  4737  rnoprab  5506
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