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Theorem rnpropg 5533
 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 4128 . . 3 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
21rneqi 5273 . 2 ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
3 rnsnopg 5532 . . . . 5 (𝐴𝑉 → ran {⟨𝐴, 𝐶⟩} = {𝐶})
43adantr 480 . . . 4 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩} = {𝐶})
5 rnsnopg 5532 . . . . 5 (𝐵𝑊 → ran {⟨𝐵, 𝐷⟩} = {𝐷})
65adantl 481 . . . 4 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐵, 𝐷⟩} = {𝐷})
74, 6uneq12d 3730 . . 3 ((𝐴𝑉𝐵𝑊) → (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷}))
8 rnun 5460 . . 3 ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
9 df-pr 4128 . . 3 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
107, 8, 93eqtr4g 2669 . 2 ((𝐴𝑉𝐵𝑊) → ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷})
112, 10syl5eq 2656 1 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  {csn 4125  {cpr 4127  ⟨cop 4131  ran crn 5039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049 This theorem is referenced by:  funcnvtp  5865  funcnvqp  5866  funcnvqpOLD  5867  esumsnf  29453  poimirlem9  32588  sge0sn  39272  umgr2v2eedg  40740
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