Theorem 1stval 8004

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 4590	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	dmeqd 5886	. . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
3	2	unieqd 4879	. . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴})
4		df-1st 8002	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5		snex 5396	. . . . 5 ⊢ {𝐴} ∈ V
6	5	dmex 7922	. . . 4 ⊢ dom {𝐴} ∈ V
7	6	uniex 7759	. . 3 ⊢ ∪ dom {𝐴} ∈ V
8	3, 4, 7	fvmpt 6994	. 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
9		fvprc 6878	. . 3 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∅)
10		snprc 4674	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 219	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	dmeqd 5886	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = dom ∅)
13		dm0 5901	. . . . . 6 ⊢ dom ∅ = ∅
14	12, 13	eqtrdi 2811	. . . . 5 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = ∅)
15	14	unieqd 4879	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∪ ∅)
16		uni0 4895	. . . 4 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2811	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∅)
18	9, 17	eqtr4d 2798	. 2 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
19	8, 18	pm2.61i 184	1 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ∅c0 4278  {csn 4584  ∪ cuni 4866  dom cdm 5650  ‘cfv 6539  1^st c1st 8000

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-iota 6494  df-fun 6541  df-fv 6547  df-1st 8002

This theorem is used by:  1stnpr  8006  1st0  8008  op1st  8010  1st2val  8030  elxp6  8036  mpoxopxnop0  8236