Theorem 1stval 7973

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 4633	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	dmeqd 5898	. . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
3	2	unieqd 4915	. . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴})
4		df-1st 7971	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5		snex 5424	. . . . 5 ⊢ {𝐴} ∈ V
6	5	dmex 7898	. . . 4 ⊢ dom {𝐴} ∈ V
7	6	uniex 7727	. . 3 ⊢ ∪ dom {𝐴} ∈ V
8	3, 4, 7	fvmpt 6991	. 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
9		fvprc 6876	. . 3 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∅)
10		snprc 4716	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 215	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	dmeqd 5898	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = dom ∅)
13		dm0 5913	. . . . . 6 ⊢ dom ∅ = ∅
14	12, 13	eqtrdi 2782	. . . . 5 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = ∅)
15	14	unieqd 4915	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∪ ∅)
16		uni0 4932	. . . 4 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2782	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∅)
18	9, 17	eqtr4d 2769	. 2 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
19	8, 18	pm2.61i 182	1 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098  Vcvv 3468  ∅c0 4317  {csn 4623  ∪ cuni 4902  dom cdm 5669  ‘cfv 6536  1^st c1st 7969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fv 6544  df-1st 7971

This theorem is referenced by:  1stnpr  7975  1st0  7977  op1st  7979  1st2val  7999  elxp6  8005  mpoxopxnop0  8198