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Theorem 2ndval 7701
 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 4535 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21rneqd 5783 . . . 4 (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
32unieqd 4815 . . 3 (𝑥 = 𝐴 ran {𝑥} = ran {𝐴})
4 df-2nd 7699 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
5 snex 5303 . . . . 5 {𝐴} ∈ V
65rnex 7627 . . . 4 ran {𝐴} ∈ V
76uniex 7470 . . 3 ran {𝐴} ∈ V
83, 4, 7fvmpt 6763 . 2 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
9 fvprc 6654 . . 3 𝐴 ∈ V → (2nd𝐴) = ∅)
10 snprc 4613 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 219 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211rneqd 5783 . . . . . 6 𝐴 ∈ V → ran {𝐴} = ran ∅)
13 rn0 5771 . . . . . 6 ran ∅ = ∅
1412, 13eqtrdi 2809 . . . . 5 𝐴 ∈ V → ran {𝐴} = ∅)
1514unieqd 4815 . . . 4 𝐴 ∈ V → ran {𝐴} = ∅)
16 uni0 4831 . . . 4 ∅ = ∅
1715, 16eqtrdi 2809 . . 3 𝐴 ∈ V → ran {𝐴} = ∅)
189, 17eqtr4d 2796 . 2 𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
198, 18pm2.61i 185 1 (2nd𝐴) = ran {𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ∅c0 4227  {csn 4525  ∪ cuni 4801  ran crn 5528  ‘cfv 6339  2nd c2nd 7697 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-iota 6298  df-fun 6341  df-fv 6347  df-2nd 7699 This theorem is referenced by:  2ndnpr  7703  2nd0  7705  op2nd  7707  2nd2val  7727  elxp6  7732
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