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Theorem 1stval 8001
Description: The value of the function that extracts the first member of an ordered pair. (Contributed by NM, 9-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
1stval (1st𝐴) = dom {𝐴}

Proof of Theorem 1stval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 4642 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21dmeqd 5912 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
32unieqd 4925 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
4 df-1st 7999 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
5 snex 5437 . . . . 5 {𝐴} ∈ V
65dmex 7923 . . . 4 dom {𝐴} ∈ V
76uniex 7752 . . 3 dom {𝐴} ∈ V
83, 4, 7fvmpt 7010 . 2 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
9 fvprc 6894 . . 3 𝐴 ∈ V → (1st𝐴) = ∅)
10 snprc 4726 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 215 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211dmeqd 5912 . . . . . 6 𝐴 ∈ V → dom {𝐴} = dom ∅)
13 dm0 5927 . . . . . 6 dom ∅ = ∅
1412, 13eqtrdi 2784 . . . . 5 𝐴 ∈ V → dom {𝐴} = ∅)
1514unieqd 4925 . . . 4 𝐴 ∈ V → dom {𝐴} = ∅)
16 uni0 4942 . . . 4 ∅ = ∅
1715, 16eqtrdi 2784 . . 3 𝐴 ∈ V → dom {𝐴} = ∅)
189, 17eqtr4d 2771 . 2 𝐴 ∈ V → (1st𝐴) = dom {𝐴})
198, 18pm2.61i 182 1 (1st𝐴) = dom {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3473  c0 4326  {csn 4632   cuni 4912  dom cdm 5682  cfv 6553  1st c1st 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-1st 7999
This theorem is referenced by:  1stnpr  8003  1st0  8005  op1st  8007  1st2val  8027  elxp6  8033  mpoxopxnop0  8227
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