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Theorem 1stval 7550
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 4484 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21dmeqd 5663 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
32unieqd 4757 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
4 df-1st 7548 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
5 snex 5226 . . . . 5 {𝐴} ∈ V
65dmex 7475 . . . 4 dom {𝐴} ∈ V
76uniex 7326 . . 3 dom {𝐴} ∈ V
83, 4, 7fvmpt 6638 . 2 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
9 fvprc 6534 . . 3 𝐴 ∈ V → (1st𝐴) = ∅)
10 snprc 4562 . . . . . . . 8 𝐴 ∈ V ↔ {𝐴} = ∅)
1110biimpi 217 . . . . . . 7 𝐴 ∈ V → {𝐴} = ∅)
1211dmeqd 5663 . . . . . 6 𝐴 ∈ V → dom {𝐴} = dom ∅)
13 dm0 5679 . . . . . 6 dom ∅ = ∅
1412, 13syl6eq 2846 . . . . 5 𝐴 ∈ V → dom {𝐴} = ∅)
1514unieqd 4757 . . . 4 𝐴 ∈ V → dom {𝐴} = ∅)
16 uni0 4774 . . . 4 ∅ = ∅
1715, 16syl6eq 2846 . . 3 𝐴 ∈ V → dom {𝐴} = ∅)
189, 17eqtr4d 2833 . 2 𝐴 ∈ V → (1st𝐴) = dom {𝐴})
198, 18pm2.61i 183 1 (1st𝐴) = dom {𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1522  wcel 2080  Vcvv 3436  c0 4213  {csn 4474   cuni 4747  dom cdm 5446  cfv 6228  1st c1st 7546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-10 2111  ax-11 2125  ax-12 2140  ax-13 2343  ax-ext 2768  ax-sep 5097  ax-nul 5104  ax-pow 5160  ax-pr 5224  ax-un 7322
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1763  df-nf 1767  df-sb 2042  df-mo 2575  df-eu 2611  df-clab 2775  df-cleq 2787  df-clel 2862  df-nfc 2934  df-ral 3109  df-rex 3110  df-rab 3113  df-v 3438  df-sbc 3708  df-dif 3864  df-un 3866  df-in 3868  df-ss 3876  df-nul 4214  df-if 4384  df-sn 4475  df-pr 4477  df-op 4481  df-uni 4748  df-br 4965  df-opab 5027  df-mpt 5044  df-id 5351  df-xp 5452  df-rel 5453  df-cnv 5454  df-co 5455  df-dm 5456  df-rn 5457  df-iota 6192  df-fun 6230  df-fv 6236  df-1st 7548
This theorem is referenced by:  1stnpr  7552  1st0  7554  op1st  7556  1st2val  7576  elxp6  7582  mpoxopxnop0  7735
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