Theorem 1stval 7989

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 4600	. . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	dmeqd 5897	. . . 4 ⊢ (𝑥 = 𝐴 → dom {𝑥} = dom {𝐴})
3	2	unieqd 4886	. . 3 ⊢ (𝑥 = 𝐴 → ∪ dom {𝑥} = ∪ dom {𝐴})
4		df-1st 7987	. . 3 ⊢ 1st = (𝑥 ∈ V ↦ ∪ dom {𝑥})
5		snex 5412	. . . . 5 ⊢ {𝐴} ∈ V
6	5	dmex 7907	. . . 4 ⊢ dom {𝐴} ∈ V
7	6	uniex 7741	. . 3 ⊢ ∪ dom {𝐴} ∈ V
8	3, 4, 7	fvmpt 6991	. 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
9		fvprc 6875	. . 3 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∅)
10		snprc 4684	. . . . . . . 8 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
11	10	biimpi 219	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
12	11	dmeqd 5897	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = dom ∅)
13		dm0 5912	. . . . . 6 ⊢ dom ∅ = ∅
14	12, 13	eqtrdi 2814	. . . . 5 ⊢ (¬ 𝐴 ∈ V → dom {𝐴} = ∅)
15	14	unieqd 4886	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∪ ∅)
16		uni0 4902	. . . 4 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2814	. . 3 ⊢ (¬ 𝐴 ∈ V → ∪ dom {𝐴} = ∅)
18	9, 17	eqtr4d 2801	. 2 ⊢ (¬ 𝐴 ∈ V → (1st ‘𝐴) = ∪ dom {𝐴})
19	8, 18	pm2.61i 184	1 ⊢ (1st ‘𝐴) = ∪ dom {𝐴}

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455  ∅c0 4287  {csn 4590  ∪ cuni 4873  dom cdm 5663  ‘cfv 6538  1^st c1st 7985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fv 6546  df-1st 7987

This theorem is referenced by:  1stnpr  7991  1st0  7993  op1st  7995  1st2val  8015  elxp6  8021  mpoxopxnop0  8212