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 → (1st ‘𝐴) = ∪ dom {𝐴})

Proof of Theorem 1stvalg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	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)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝐴 ∈ V → ∪ dom {𝐴} ∈ V)
5		sneq 3433	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
6	5	dmeqd 4596	. . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
7	6	unieqd 3638	. . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴})
8		df-1st 5846	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
9	7, 8	fvmptg 5325	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ dom {𝐴} ∈ V) → (1st ‘𝐴) = ∪ dom {𝐴})
10	4, 9	mpdan 412	1 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})

Syntax hints:   → wi 4   = wceq 1285   ∈ wcel 1434  Vcvv 2612  {csn 3422  ∪ cuni 3627  dom cdm 4401  ‘cfv 4969  1^st 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