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Theorem 2ndval 7062
 Description: The value of the function that extracts the second member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
2ndval (2nd𝐴) = ran {𝐴}

Proof of Theorem 2ndval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 4135 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21rneqd 5274 . . . 4 (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
32unieqd 4382 . . 3 (𝑥 = 𝐴 ran {𝑥} = ran {𝐴})
4 df-2nd 7060 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
5 snex 4835 . . . . 5 {𝐴} ∈ V
65rnex 6992 . . . 4 ran {𝐴} ∈ V
76uniex 6851 . . 3 ran {𝐴} ∈ V
83, 4, 7fvmpt 6191 . 2 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
9 fvprc 6097 . . 3 𝐴 ∈ V → (2nd𝐴) = ∅)
10 snprc 4197 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 205 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211rneqd 5274 . . . . . 6 𝐴 ∈ V → ran {𝐴} = ran ∅)
13 rn0 5298 . . . . . 6 ran ∅ = ∅
1412, 13syl6eq 2660 . . . . 5 𝐴 ∈ V → ran {𝐴} = ∅)
1514unieqd 4382 . . . 4 𝐴 ∈ V → ran {𝐴} = ∅)
16 uni0 4401 . . . 4 ∅ = ∅
1715, 16syl6eq 2660 . . 3 𝐴 ∈ V → ran {𝐴} = ∅)
189, 17eqtr4d 2647 . 2 𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
198, 18pm2.61i 175 1 (2nd𝐴) = ran {𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  ∪ cuni 4372  ran crn 5039  ‘cfv 5804  2nd c2nd 7058 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-8 1979  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-pow 4769  ax-pr 4833  ax-un 6847 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-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  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-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-2nd 7060 This theorem is referenced by:  2ndnpr  7064  2nd0  7066  op2nd  7068  2nd2val  7086  elxp6  7091
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