MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnpropg Structured version   Visualization version   GIF version

Theorem rnpropg 6231
Description: The range of a pair of ordered pairs is the pair of second members. (Contributed by Thierry Arnoux, 3-Jan-2017.)
Assertion
Ref Expression
rnpropg ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})

Proof of Theorem rnpropg
StepHypRef Expression
1 df-pr 4635 . . 3 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
21rneqi 5943 . 2 ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
3 rnsnopg 6230 . . . . 5 (𝐴𝑉 → ran {⟨𝐴, 𝐶⟩} = {𝐶})
43adantr 479 . . . 4 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩} = {𝐶})
5 rnsnopg 6230 . . . . 5 (𝐵𝑊 → ran {⟨𝐵, 𝐷⟩} = {𝐷})
65adantl 480 . . . 4 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐵, 𝐷⟩} = {𝐷})
74, 6uneq12d 4165 . . 3 ((𝐴𝑉𝐵𝑊) → (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷}))
8 rnun 6155 . . 3 ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
9 df-pr 4635 . . 3 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
107, 8, 93eqtr4g 2793 . 2 ((𝐴𝑉𝐵𝑊) → ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷})
112, 10eqtrid 2780 1 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  cun 3947  {csn 4632  {cpr 4634  cop 4638  ran crn 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693
This theorem is referenced by:  funcnvtp  6621  funcnvqp  6622  umgr2v2eedg  29358  esumsnf  33716  poimirlem9  37135  sge0sn  45796
  Copyright terms: Public domain W3C validator