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Theorem 2ndval 7978
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 4639 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21rneqd 5938 . . . 4 (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
32unieqd 4923 . . 3 (𝑥 = 𝐴 ran {𝑥} = ran {𝐴})
4 df-2nd 7976 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
5 snex 5432 . . . . 5 {𝐴} ∈ V
65rnex 7903 . . . 4 ran {𝐴} ∈ V
76uniex 7731 . . 3 ran {𝐴} ∈ V
83, 4, 7fvmpt 6999 . 2 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
9 fvprc 6884 . . 3 𝐴 ∈ V → (2nd𝐴) = ∅)
10 snprc 4722 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 215 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211rneqd 5938 . . . . . 6 𝐴 ∈ V → ran {𝐴} = ran ∅)
13 rn0 5926 . . . . . 6 ran ∅ = ∅
1412, 13eqtrdi 2789 . . . . 5 𝐴 ∈ V → ran {𝐴} = ∅)
1514unieqd 4923 . . . 4 𝐴 ∈ V → ran {𝐴} = ∅)
16 uni0 4940 . . . 4 ∅ = ∅
1715, 16eqtrdi 2789 . . 3 𝐴 ∈ V → ran {𝐴} = ∅)
189, 17eqtr4d 2776 . 2 𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
198, 18pm2.61i 182 1 (2nd𝐴) = ran {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  c0 4323  {csn 4629   cuni 4909  ran crn 5678  cfv 6544  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fv 6552  df-2nd 7976
This theorem is referenced by:  2ndnpr  7980  2nd0  7982  op2nd  7984  2nd2val  8004  elxp6  8009
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