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Theorem 2ndval 7336
 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 4331 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21rneqd 5508 . . . 4 (𝑥 = 𝐴 → ran {𝑥} = ran {𝐴})
32unieqd 4598 . . 3 (𝑥 = 𝐴 ran {𝑥} = ran {𝐴})
4 df-2nd 7334 . . 3 2nd = (𝑥 ∈ V ↦ ran {𝑥})
5 snex 5057 . . . . 5 {𝐴} ∈ V
65rnex 7265 . . . 4 ran {𝐴} ∈ V
76uniex 7118 . . 3 ran {𝐴} ∈ V
83, 4, 7fvmpt 6444 . 2 (𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
9 fvprc 6346 . . 3 𝐴 ∈ V → (2nd𝐴) = ∅)
10 snprc 4397 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 206 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211rneqd 5508 . . . . . 6 𝐴 ∈ V → ran {𝐴} = ran ∅)
13 rn0 5532 . . . . . 6 ran ∅ = ∅
1412, 13syl6eq 2810 . . . . 5 𝐴 ∈ V → ran {𝐴} = ∅)
1514unieqd 4598 . . . 4 𝐴 ∈ V → ran {𝐴} = ∅)
16 uni0 4617 . . . 4 ∅ = ∅
1715, 16syl6eq 2810 . . 3 𝐴 ∈ V → ran {𝐴} = ∅)
189, 17eqtr4d 2797 . 2 𝐴 ∈ V → (2nd𝐴) = ran {𝐴})
198, 18pm2.61i 176 1 (2nd𝐴) = ran {𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ∅c0 4058  {csn 4321  ∪ cuni 4588  ran crn 5267  ‘cfv 6049  2nd c2nd 7332 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-2nd 7334 This theorem is referenced by:  2ndnpr  7338  2nd0  7340  op2nd  7342  2nd2val  7362  elxp6  7367
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