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Theorem 2ndval 7925
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 4597 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21rneqd 5894 . . . 4 (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
32unieqd 4880 . . 3 (𝑥 = 𝐴 ran {𝑥} = ran {𝐴})
4 df-2nd 7923 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
5 snex 5389 . . . . 5 {𝐴} ∈ V
65rnex 7850 . . . 4 ran {𝐴} ∈ V
76uniex 7679 . . 3 ran {𝐴} ∈ V
83, 4, 7fvmpt 6949 . 2 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
9 fvprc 6835 . . 3 𝐴 ∈ V → (2nd𝐴) = ∅)
10 snprc 4679 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 215 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211rneqd 5894 . . . . . 6 𝐴 ∈ V → ran {𝐴} = ran ∅)
13 rn0 5882 . . . . . 6 ran ∅ = ∅
1412, 13eqtrdi 2793 . . . . 5 𝐴 ∈ V → ran {𝐴} = ∅)
1514unieqd 4880 . . . 4 𝐴 ∈ V → ran {𝐴} = ∅)
16 uni0 4897 . . . 4 ∅ = ∅
1715, 16eqtrdi 2793 . . 3 𝐴 ∈ V → ran {𝐴} = ∅)
189, 17eqtr4d 2780 . 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 3446  c0 4283  {csn 4587   cuni 4866  ran crn 5635  cfv 6497  2nd c2nd 7921
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 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fv 6505  df-2nd 7923
This theorem is referenced by:  2ndnpr  7927  2nd0  7929  op2nd  7931  2nd2val  7951  elxp6  7956
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