Theorem 1stval 7833

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 4571	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	dmeqd 5814	. . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
3	2	unieqd 4853	. . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴})
4		df-1st 7831	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5		snex 5354	. . . . 5 ⊢ {𝐴} ∈ V
6	5	dmex 7758	. . . 4 ⊢ dom {𝐴} ∈ V
7	6	uniex 7594	. . 3 ⊢ ∪ dom {𝐴} ∈ V
8	3, 4, 7	fvmpt 6875	. 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
9		fvprc 6766	. . 3 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∅)
10		snprc 4653	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 215	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	dmeqd 5814	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = dom ∅)
13		dm0 5829	. . . . . 6 ⊢ dom ∅ = ∅
14	12, 13	eqtrdi 2794	. . . . 5 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = ∅)
15	14	unieqd 4853	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∪ ∅)
16		uni0 4869	. . . 4 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2794	. . 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 2106  Vcvv 3432  ∅c0 4256  {csn 4561  ∪ cuni 4839  dom cdm 5589  ‘cfv 6433  1^st c1st 7829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fv 6441  df-1st 7831

This theorem is referenced by:  1stnpr  7835  1st0  7837  op1st  7839  1st2val  7859  elxp6  7865  mpoxopxnop0  8031