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Theorem 1stvalg 6040
 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 4108 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
2 dmexg 4803 . . 3 ({𝐴} ∈ V → dom {𝐴} ∈ V)
3 uniexg 4361 . . 3 (dom {𝐴} ∈ V → dom {𝐴} ∈ V)
41, 2, 33syl 17 . 2 (𝐴 ∈ V → dom {𝐴} ∈ V)
5 sneq 3538 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
65dmeqd 4741 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
76unieqd 3747 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
8 df-1st 6038 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
97, 8fvmptg 5497 . 2 ((𝐴 ∈ V ∧ dom {𝐴} ∈ V) → (1st𝐴) = dom {𝐴})
104, 9mpdan 417 1 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1331   ∈ wcel 1480  Vcvv 2686  {csn 3527  ∪ cuni 3736  dom cdm 4539  ‘cfv 5123  1st c1st 6036 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fv 5131  df-1st 6038 This theorem is referenced by:  1st0  6042  op1st  6044  elxp6  6067
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