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Theorem 2ndval 7683
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 4574 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21rneqd 5807 . . . 4 (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
32unieqd 4847 . . 3 (𝑥 = 𝐴 ran {𝑥} = ran {𝐴})
4 df-2nd 7681 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
5 snex 5328 . . . . 5 {𝐴} ∈ V
65rnex 7605 . . . 4 ran {𝐴} ∈ V
76uniex 7455 . . 3 ran {𝐴} ∈ V
83, 4, 7fvmpt 6765 . 2 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
9 fvprc 6660 . . 3 𝐴 ∈ V → (2nd𝐴) = ∅)
10 snprc 4652 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 217 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211rneqd 5807 . . . . . 6 𝐴 ∈ V → ran {𝐴} = ran ∅)
13 rn0 5795 . . . . . 6 ran ∅ = ∅
1412, 13syl6eq 2877 . . . . 5 𝐴 ∈ V → ran {𝐴} = ∅)
1514unieqd 4847 . . . 4 𝐴 ∈ V → ran {𝐴} = ∅)
16 uni0 4864 . . . 4 ∅ = ∅
1715, 16syl6eq 2877 . . 3 𝐴 ∈ V → ran {𝐴} = ∅)
189, 17eqtr4d 2864 . 2 𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
198, 18pm2.61i 183 1 (2nd𝐴) = ran {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  wcel 2107  Vcvv 3500  c0 4295  {csn 4564   cuni 4837  ran crn 5555  cfv 6352  2nd c2nd 7679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6312  df-fun 6354  df-fv 6360  df-2nd 7681
This theorem is referenced by:  2ndnpr  7685  2nd0  7687  op2nd  7689  2nd2val  7709  elxp6  7714
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