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Theorem 2ndvalg 5847
 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
2ndvalg (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})

Proof of Theorem 2ndvalg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 snexg 3982 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
2 rnexg 4654 . . 3 ({𝐴} ∈ V → ran {𝐴} ∈ V)
3 uniexg 4228 . . 3 (ran {𝐴} ∈ V → ran {𝐴} ∈ V)
41, 2, 33syl 17 . 2 (𝐴 ∈ V → ran {𝐴} ∈ V)
5 sneq 3433 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
65rneqd 4620 . . . 4 (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
76unieqd 3638 . . 3 (𝑥 = 𝐴 ran {𝑥} = ran {𝐴})
8 df-2nd 5845 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
97, 8fvmptg 5323 . 2 ((𝐴 ∈ V ∧ ran {𝐴} ∈ V) → (2nd𝐴) = ran {𝐴})
104, 9mpdan 412 1 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1285   ∈ wcel 1434  Vcvv 2612  {csn 3422  ∪ cuni 3627  ran crn 4400  ‘cfv 4967  2nd c2nd 5843 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999  ax-un 4223 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4083  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-dm 4409  df-rn 4410  df-iota 4932  df-fun 4969  df-fv 4975  df-2nd 5845 This theorem is referenced by:  2nd0  5849  op2nd  5851  elxp6  5873
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