Theorem 1stvalg 6376

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 4321	. . 3 ⊢ (𝐴 ∈ V → {𝐴} ∈ V)
2		dmexg 5046	. . 3 ⊢ ({𝐴} ∈ V → dom {𝐴} ∈ V)
3		uniexg 4585	. . 3 ⊢ (dom {𝐴} ∈ V → ∪ dom {𝐴} ∈ V)
4	1, 2, 3	3syl 17	. 2 ⊢ (𝐴 ∈ V → ∪ dom {𝐴} ∈ V)
5		sneq 3720	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
6	5	dmeqd 4983	. . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
7	6	unieqd 3946	. . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴})
8		df-1st 6374	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
9	7, 8	fvmptg 5781	. 2 ⊢ ((𝐴 ∈ V ∧ ∪ dom {𝐴} ∈ V) → (1st ‘𝐴) = ∪ dom {𝐴})
10	4, 9	mpdan 425	1 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})

This proof depends on syntax axioms:   → wi 4   = wceq 1402   ∈ wcel 2209  Vcvv 2821  {csn 3709  ∪ cuni 3935  dom cdm 4774  ‘cfv 5377  1^st c1st 6372

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fv 5385  df-1st 6374

This theorem is used by:  1st0  6378  op1st  6380  elxp6  6403