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Theorem 1stvalg 5848
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 3983 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
2 dmexg 4655 . . 3 ({𝐴} ∈ V → dom {𝐴} ∈ V)
3 uniexg 4229 . . 3 (dom {𝐴} ∈ V → dom {𝐴} ∈ V)
41, 2, 33syl 17 . 2 (𝐴 ∈ V → dom {𝐴} ∈ V)
5 sneq 3433 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
65dmeqd 4596 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
76unieqd 3638 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
8 df-1st 5846 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
97, 8fvmptg 5325 . 2 ((𝐴 ∈ V ∧ dom {𝐴} ∈ V) → (1st𝐴) = dom {𝐴})
104, 9mpdan 412 1 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  wcel 1434  Vcvv 2612  {csn 3422   cuni 3627  dom cdm 4401  cfv 4969  1st c1st 5844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-iota 4934  df-fun 4971  df-fv 4977  df-1st 5846
This theorem is referenced by:  1st0  5850  op1st  5852  elxp6  5875
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