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Theorem 2ndval 8015
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 4640 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21rneqd 5951 . . . 4 (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
32unieqd 4924 . . 3 (𝑥 = 𝐴 ran {𝑥} = ran {𝐴})
4 df-2nd 8013 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
5 snex 5441 . . . . 5 {𝐴} ∈ V
65rnex 7932 . . . 4 ran {𝐴} ∈ V
76uniex 7759 . . 3 ran {𝐴} ∈ V
83, 4, 7fvmpt 7015 . 2 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
9 fvprc 6898 . . 3 𝐴 ∈ V → (2nd𝐴) = ∅)
10 snprc 4721 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 216 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211rneqd 5951 . . . . . 6 𝐴 ∈ V → ran {𝐴} = ran ∅)
13 rn0 5938 . . . . . 6 ran ∅ = ∅
1412, 13eqtrdi 2790 . . . . 5 𝐴 ∈ V → ran {𝐴} = ∅)
1514unieqd 4924 . . . 4 𝐴 ∈ V → ran {𝐴} = ∅)
16 uni0 4939 . . . 4 ∅ = ∅
1715, 16eqtrdi 2790 . . 3 𝐴 ∈ V → ran {𝐴} = ∅)
189, 17eqtr4d 2777 . 2 𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
198, 18pm2.61i 182 1 (2nd𝐴) = ran {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1536  wcel 2105  Vcvv 3477  c0 4338  {csn 4630   cuni 4911  ran crn 5689  cfv 6562  2nd c2nd 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fv 6570  df-2nd 8013
This theorem is referenced by:  2ndnpr  8017  2nd0  8019  op2nd  8021  2nd2val  8041  elxp6  8046
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