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Theorem 2ndvalg 5712
 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 snexgOLD 3926 . . 3 (A V → {A} V)
2 rnexg 4540 . . 3 ({A} V → ran {A} V)
3 uniexg 4141 . . 3 (ran {A} V → ran {A} V)
41, 2, 33syl 17 . 2 (A V → ran {A} V)
5 sneq 3378 . . . . 5 (x = A → {x} = {A})
65rneqd 4506 . . . 4 (x = A → ran {x} = ran {A})
76unieqd 3582 . . 3 (x = A ran {x} = ran {A})
8 df-2nd 5710 . . 3 2nd = (x V ↦ ran {x})
97, 8fvmptg 5191 . 2 ((A V ran {A} V) → (2ndA) = ran {A})
104, 9mpdan 398 1 (A V → (2ndA) = ran {A})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242   ∈ wcel 1390  Vcvv 2551  {csn 3367  ∪ cuni 3571  ran crn 4289  ‘cfv 4845  2nd c2nd 5708 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-2nd 5710 This theorem is referenced by:  2nd0  5714  op2nd  5716  elxp6  5738
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