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Theorem 1stvalg 6162
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
1stvalg (𝐴 ∈ V → (1st𝐴) = dom {𝐴})

Proof of Theorem 1stvalg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 snexg 4199 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
2 dmexg 4906 . . 3 ({𝐴} ∈ V → dom {𝐴} ∈ V)
3 uniexg 4454 . . 3 (dom {𝐴} ∈ V → dom {𝐴} ∈ V)
41, 2, 33syl 17 . 2 (𝐴 ∈ V → dom {𝐴} ∈ V)
5 sneq 3618 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
65dmeqd 4844 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
76unieqd 3835 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
8 df-1st 6160 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
97, 8fvmptg 5609 . 2 ((𝐴 ∈ V ∧ dom {𝐴} ∈ V) → (1st𝐴) = dom {𝐴})
104, 9mpdan 421 1 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wcel 2160  Vcvv 2752  {csn 3607   cuni 3824  dom cdm 4641  cfv 5232  1st c1st 6158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-iota 5193  df-fun 5234  df-fv 5240  df-1st 6160
This theorem is referenced by:  1st0  6164  op1st  6166  elxp6  6189
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