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Theorem 2ndvalg 6034
 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 4103 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
2 rnexg 4799 . . 3 ({𝐴} ∈ V → ran {𝐴} ∈ V)
3 uniexg 4356 . . 3 (ran {𝐴} ∈ V → ran {𝐴} ∈ V)
41, 2, 33syl 17 . 2 (𝐴 ∈ V → ran {𝐴} ∈ V)
5 sneq 3533 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
65rneqd 4763 . . . 4 (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
76unieqd 3742 . . 3 (𝑥 = 𝐴 ran {𝑥} = ran {𝐴})
8 df-2nd 6032 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
97, 8fvmptg 5490 . 2 ((𝐴 ∈ V ∧ ran {𝐴} ∈ V) → (2nd𝐴) = ran {𝐴})
104, 9mpdan 417 1 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  Vcvv 2681  {csn 3522  ∪ cuni 3731  ran crn 4535  ‘cfv 5118  2nd c2nd 6030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-iota 5083  df-fun 5120  df-fv 5126  df-2nd 6032 This theorem is referenced by:  2nd0  6036  op2nd  6038  elxp6  6060
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