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Theorem 1stval 7973
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 4637 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21dmeqd 5903 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
32unieqd 4921 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
4 df-1st 7971 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
5 snex 5430 . . . . 5 {𝐴} ∈ V
65dmex 7898 . . . 4 dom {𝐴} ∈ V
76uniex 7727 . . 3 dom {𝐴} ∈ V
83, 4, 7fvmpt 6995 . 2 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
9 fvprc 6880 . . 3 𝐴 ∈ V → (1st𝐴) = ∅)
10 snprc 4720 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 215 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211dmeqd 5903 . . . . . 6 𝐴 ∈ V → dom {𝐴} = dom ∅)
13 dm0 5918 . . . . . 6 dom ∅ = ∅
1412, 13eqtrdi 2788 . . . . 5 𝐴 ∈ V → dom {𝐴} = ∅)
1514unieqd 4921 . . . 4 𝐴 ∈ V → dom {𝐴} = ∅)
16 uni0 4938 . . . 4 ∅ = ∅
1715, 16eqtrdi 2788 . . 3 𝐴 ∈ V → dom {𝐴} = ∅)
189, 17eqtr4d 2775 . 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 1541  wcel 2106  Vcvv 3474  c0 4321  {csn 4627   cuni 4907  dom cdm 5675  cfv 6540  1st c1st 7969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-1st 7971
This theorem is referenced by:  1stnpr  7975  1st0  7977  op1st  7979  1st2val  7999  elxp6  8005  mpoxopxnop0  8196
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