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Theorem 1stval 7675
 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 4535 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21dmeqd 5738 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
32unieqd 4814 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
4 df-1st 7673 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
5 snex 5297 . . . . 5 {𝐴} ∈ V
65dmex 7600 . . . 4 dom {𝐴} ∈ V
76uniex 7449 . . 3 dom {𝐴} ∈ V
83, 4, 7fvmpt 6745 . 2 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
9 fvprc 6638 . . 3 𝐴 ∈ V → (1st𝐴) = ∅)
10 snprc 4613 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 219 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211dmeqd 5738 . . . . . 6 𝐴 ∈ V → dom {𝐴} = dom ∅)
13 dm0 5754 . . . . . 6 dom ∅ = ∅
1412, 13eqtrdi 2849 . . . . 5 𝐴 ∈ V → dom {𝐴} = ∅)
1514unieqd 4814 . . . 4 𝐴 ∈ V → dom {𝐴} = ∅)
16 uni0 4828 . . . 4 ∅ = ∅
1715, 16eqtrdi 2849 . . 3 𝐴 ∈ V → dom {𝐴} = ∅)
189, 17eqtr4d 2836 . 2 𝐴 ∈ V → (1st𝐴) = dom {𝐴})
198, 18pm2.61i 185 1 (1st𝐴) = dom {𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  {csn 4525  ∪ cuni 4800  dom cdm 5519  ‘cfv 6324  1st c1st 7671 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fv 6332  df-1st 7673 This theorem is referenced by:  1stnpr  7677  1st0  7679  op1st  7681  1st2val  7701  elxp6  7707  mpoxopxnop0  7866
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