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Theorem 1stval 7212
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 4220 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21dmeqd 5358 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
32unieqd 4478 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
4 df-1st 7210 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
5 snex 4938 . . . . 5 {𝐴} ∈ V
65dmex 7141 . . . 4 dom {𝐴} ∈ V
76uniex 6995 . . 3 dom {𝐴} ∈ V
83, 4, 7fvmpt 6321 . 2 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
9 fvprc 6223 . . 3 𝐴 ∈ V → (1st𝐴) = ∅)
10 snprc 4285 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 206 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211dmeqd 5358 . . . . . 6 𝐴 ∈ V → dom {𝐴} = dom ∅)
13 dm0 5371 . . . . . 6 dom ∅ = ∅
1412, 13syl6eq 2701 . . . . 5 𝐴 ∈ V → dom {𝐴} = ∅)
1514unieqd 4478 . . . 4 𝐴 ∈ V → dom {𝐴} = ∅)
16 uni0 4497 . . . 4 ∅ = ∅
1715, 16syl6eq 2701 . . 3 𝐴 ∈ V → dom {𝐴} = ∅)
189, 17eqtr4d 2688 . 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 1523  wcel 2030  Vcvv 3231  c0 3948  {csn 4210   cuni 4468  dom cdm 5143  cfv 5926  1st c1st 7208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-1st 7210
This theorem is referenced by:  1stnpr  7214  1st0  7216  op1st  7218  1st2val  7238  elxp6  7244  mpt2xopxnop0  7386
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