Theorem 1stval 7806

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
1stval	⊢ (1st ‘𝐴) = ∪ dom {𝐴}

Proof of Theorem 1stval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4568	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	dmeqd 5803	. . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
3	2	unieqd 4850	. . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴})
4		df-1st 7804	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5		snex 5349	. . . . 5 ⊢ {𝐴} ∈ V
6	5	dmex 7732	. . . 4 ⊢ dom {𝐴} ∈ V
7	6	uniex 7572	. . 3 ⊢ ∪ dom {𝐴} ∈ V
8	3, 4, 7	fvmpt 6857	. 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
9		fvprc 6748	. . 3 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∅)
10		snprc 4650	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 215	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	dmeqd 5803	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = dom ∅)
13		dm0 5818	. . . . . 6 ⊢ dom ∅ = ∅
14	12, 13	eqtrdi 2795	. . . . 5 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = ∅)
15	14	unieqd 4850	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∪ ∅)
16		uni0 4866	. . . 4 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2795	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∅)
18	9, 17	eqtr4d 2781	. 2 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
19	8, 18	pm2.61i 182	1 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108  Vcvv 3422  ∅c0 4253  {csn 4558  ∪ cuni 4836  dom cdm 5580  ‘cfv 6418  1^st c1st 7802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-1st 7804

This theorem is referenced by:  1stnpr  7808  1st0  7810  op1st  7812  1st2val  7832  elxp6  7838  mpoxopxnop0  8002