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Theorem 1stvalg 5951
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 4040 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
2 dmexg 4729 . . 3 ({𝐴} ∈ V → dom {𝐴} ∈ V)
3 uniexg 4290 . . 3 (dom {𝐴} ∈ V → dom {𝐴} ∈ V)
41, 2, 33syl 17 . 2 (𝐴 ∈ V → dom {𝐴} ∈ V)
5 sneq 3477 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
65dmeqd 4669 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
76unieqd 3686 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
8 df-1st 5949 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
97, 8fvmptg 5415 . 2 ((𝐴 ∈ V ∧ dom {𝐴} ∈ V) → (1st𝐴) = dom {𝐴})
104, 9mpdan 413 1 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1296  wcel 1445  Vcvv 2633  {csn 3466   cuni 3675  dom cdm 4467  cfv 5049  1st c1st 5947
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-sbc 2855  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-iota 5014  df-fun 5051  df-fv 5057  df-1st 5949
This theorem is referenced by:  1st0  5953  op1st  5955  elxp6  5978
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