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Theorem rnpropg 6175
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 4590 . . 3 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
21rneqi 5893 . 2 ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩})
3 rnsnopg 6174 . . . . 5 (𝐴𝑉 → ran {⟨𝐴, 𝐶⟩} = {𝐶})
43adantr 482 . . . 4 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩} = {𝐶})
5 rnsnopg 6174 . . . . 5 (𝐵𝑊 → ran {⟨𝐵, 𝐷⟩} = {𝐷})
65adantl 483 . . . 4 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐵, 𝐷⟩} = {𝐷})
74, 6uneq12d 4125 . . 3 ((𝐴𝑉𝐵𝑊) → (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩}) = ({𝐶} ∪ {𝐷}))
8 rnun 6099 . . 3 ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = (ran {⟨𝐴, 𝐶⟩} ∪ ran {⟨𝐵, 𝐷⟩})
9 df-pr 4590 . . 3 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
107, 8, 93eqtr4g 2798 . 2 ((𝐴𝑉𝐵𝑊) → ran ({⟨𝐴, 𝐶⟩} ∪ {⟨𝐵, 𝐷⟩}) = {𝐶, 𝐷})
112, 10eqtrid 2785 1 ((𝐴𝑉𝐵𝑊) → ran {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} = {𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cun 3909  {csn 4587  {cpr 4589  cop 4593  ran crn 5635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645
This theorem is referenced by:  funcnvtp  6565  funcnvqp  6566  umgr2v2eedg  28514  esumsnf  32720  poimirlem9  36133  sge0sn  44706
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