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Theorem rnpropg 6244
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 4634 . . 3 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
21rneqi 5951 . 2 ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
3 rnsnopg 6243 . . . . 5 (𝐴𝑉 → ran {⟨𝐴, 𝐶⟩} = {𝐶})
43adantr 480 . . . 4 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩} = {𝐶})
5 rnsnopg 6243 . . . . 5 (𝐵𝑊 → ran {⟨𝐵, 𝐷⟩} = {𝐷})
65adantl 481 . . . 4 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐵, 𝐷⟩} = {𝐷})
74, 6uneq12d 4179 . . 3 ((𝐴𝑉𝐵𝑊) → (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷}))
8 rnun 6168 . . 3 ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
9 df-pr 4634 . . 3 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
107, 8, 93eqtr4g 2800 . 2 ((𝐴𝑉𝐵𝑊) → ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷})
112, 10eqtrid 2787 1 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cun 3961  {csn 4631  {cpr 4633  cop 4637  ran crn 5690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700
This theorem is referenced by:  funcnvtp  6631  funcnvqp  6632  umgr2v2eedg  29557  esumsnf  34045  poimirlem9  37616  sge0sn  46335
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