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Theorem 1stvalg 5769
 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 snexgOLD 3935 . . 3 (𝐴 ∈ V → {𝐴} ∈ V)
2 dmexg 4596 . . 3 ({𝐴} ∈ V → dom {𝐴} ∈ V)
3 uniexg 4175 . . 3 (dom {𝐴} ∈ V → dom {𝐴} ∈ V)
41, 2, 33syl 17 . 2 (𝐴 ∈ V → dom {𝐴} ∈ V)
5 sneq 3386 . . . . 5 (𝑥 = 𝐴 → {𝑥} = {𝐴})
65dmeqd 4537 . . . 4 (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
76unieqd 3591 . . 3 (𝑥 = 𝐴 dom {𝑥} = dom {𝐴})
8 df-1st 5767 . . 3 1st = (𝑥 ∈ V ↦ dom {𝑥})
97, 8fvmptg 5248 . 2 ((𝐴 ∈ V ∧ dom {𝐴} ∈ V) → (1st𝐴) = dom {𝐴})
104, 9mpdan 398 1 (𝐴 ∈ V → (1st𝐴) = dom {𝐴})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375  ∪ cuni 3580  dom cdm 4345  ‘cfv 4902  1st c1st 5765 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fv 4910  df-1st 5767 This theorem is referenced by:  1st0  5771  op1st  5773  elxp6  5796
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