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Theorem 1stval 7370
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 4346 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21dmeqd 5496 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
32unieqd 4606 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
4 df-1st 7368 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
5 snex 5066 . . . . 5 {𝐴} ∈ V
65dmex 7299 . . . 4 dom {𝐴} ∈ V
76uniex 7153 . . 3 dom {𝐴} ∈ V
83, 4, 7fvmpt 6473 . 2 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
9 fvprc 6370 . . 3 𝐴 ∈ V → (1st𝐴) = ∅)
10 snprc 4410 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 207 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211dmeqd 5496 . . . . . 6 𝐴 ∈ V → dom {𝐴} = dom ∅)
13 dm0 5509 . . . . . 6 dom ∅ = ∅
1412, 13syl6eq 2815 . . . . 5 𝐴 ∈ V → dom {𝐴} = ∅)
1514unieqd 4606 . . . 4 𝐴 ∈ V → dom {𝐴} = ∅)
16 uni0 4625 . . . 4 ∅ = ∅
1715, 16syl6eq 2815 . . 3 𝐴 ∈ V → dom {𝐴} = ∅)
189, 17eqtr4d 2802 . 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 1652  wcel 2155  Vcvv 3350  c0 4081  {csn 4336   cuni 4596  dom cdm 5279  cfv 6070  1st c1st 7366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fv 6078  df-1st 7368
This theorem is referenced by:  1stnpr  7372  1st0  7374  op1st  7376  1st2val  7396  elxp6  7402  mpt2xopxnop0  7546
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