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Theorem 2ndvalg 5770
 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 3935 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
2 rnexg 4597 . . 3 ({𝐴} ∈ V → ran {𝐴} ∈ V)
3 uniexg 4175 . . 3 (ran {𝐴} ∈ V → ran {𝐴} ∈ V)
41, 2, 33syl 17 . 2 (𝐴 ∈ V → ran {𝐴} ∈ V)
5 sneq 3386 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
65rneqd 4563 . . . 4 (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
76unieqd 3591 . . 3 (𝑥 = 𝐴 ran {𝑥} = ran {𝐴})
8 df-2nd 5768 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
97, 8fvmptg 5248 . 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 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375  ∪ cuni 3580  ran crn 4346  ‘cfv 4902  2nd c2nd 5766 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-2nd 5768 This theorem is referenced by:  2nd0  5772  op2nd  5774  elxp6  5796
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