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Theorem rnpropg 5961
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 4481 . . 3 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
21rneqi 5696 . 2 ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
3 rnsnopg 5960 . . . . 5 (𝐴𝑉 → ran {⟨𝐴, 𝐶⟩} = {𝐶})
43adantr 481 . . . 4 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩} = {𝐶})
5 rnsnopg 5960 . . . . 5 (𝐵𝑊 → ran {⟨𝐵, 𝐷⟩} = {𝐷})
65adantl 482 . . . 4 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐵, 𝐷⟩} = {𝐷})
74, 6uneq12d 4067 . . 3 ((𝐴𝑉𝐵𝑊) → (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷}))
8 rnun 5887 . . 3 ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
9 df-pr 4481 . . 3 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
107, 8, 93eqtr4g 2858 . 2 ((𝐴𝑉𝐵𝑊) → ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷})
112, 10syl5eq 2845 1 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wcel 2083  cun 3863  {csn 4478  {cpr 4480  cop 4484  ran crn 5451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-xp 5456  df-rel 5457  df-cnv 5458  df-dm 5460  df-rn 5461
This theorem is referenced by:  funcnvtp  6294  funcnvqp  6295  umgr2v2eedg  26993  esumsnf  30936  poimirlem9  34453  sge0sn  42225
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