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Theorem 1stval 7316
 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 4324 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21dmeqd 5464 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
32unieqd 4582 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
4 df-1st 7314 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
5 snex 5036 . . . . 5 {𝐴} ∈ V
65dmex 7245 . . . 4 dom {𝐴} ∈ V
76uniex 7099 . . 3 dom {𝐴} ∈ V
83, 4, 7fvmpt 6424 . 2 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
9 fvprc 6326 . . 3 𝐴 ∈ V → (1st𝐴) = ∅)
10 snprc 4387 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 206 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211dmeqd 5464 . . . . . 6 𝐴 ∈ V → dom {𝐴} = dom ∅)
13 dm0 5477 . . . . . 6 dom ∅ = ∅
1412, 13syl6eq 2820 . . . . 5 𝐴 ∈ V → dom {𝐴} = ∅)
1514unieqd 4582 . . . 4 𝐴 ∈ V → dom {𝐴} = ∅)
16 uni0 4599 . . . 4 ∅ = ∅
1715, 16syl6eq 2820 . . 3 𝐴 ∈ V → dom {𝐴} = ∅)
189, 17eqtr4d 2807 . 2 𝐴 ∈ V → (1st𝐴) = dom {𝐴})
198, 18pm2.61i 176 1 (1st𝐴) = dom {𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1630   ∈ wcel 2144  Vcvv 3349  ∅c0 4061  {csn 4314  ∪ cuni 4572  dom cdm 5249  ‘cfv 6031  1st c1st 7312 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-1st 7314 This theorem is referenced by:  1stnpr  7318  1st0  7320  op1st  7322  1st2val  7342  elxp6  7348  mpt2xopxnop0  7492
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