Theorem 1stval 3089

Description: The value of the function that extracts the first member of an ordered pair. Equivalent to Definition 5.13 (i) of [Monk1] p. 52. The notation is the same as Monk's.

Assertion

Ref	Expression
1stval	⊢ (1st ‘A) = ∪dom {A}

Proof of Theorem 1stval

Step	Hyp	Ref	Expression
1		snex 1859	. . . . 5 ⊢ {A} ∈ V
2		dmexg 2551	. . . . 5 ⊢ ({A} ∈ V → dom {A} ∈ V)
3	1, 2	ax-mp 6	. . . 4 ⊢ dom {A} ∈ V
4	3	uniex 1947	. . 3 ⊢ ∪dom {A} ∈ V
5		sneq 1816	. . . . . . 7 ⊢ (x = A → {x} = {A})
6	5	dmeqd 2533	. . . . . 6 ⊢ (x = A → dom {x} = dom {A})
7	6	unieqd 1929	. . . . 5 ⊢ (x = A → ∪dom {x} = ∪dom {A})
8	7	fvopabg 2872	. . . 4 ⊢ ((A ∈ V ∧ ∪dom {A} ∈ V) → ({⟨x, y⟩∣y = ∪dom {x}} ‘A) = ∪dom {A})
9		df-1st 3087	. . . . 5 ⊢ 1st = {⟨x, y⟩∣y = ∪dom {x}}
10	9	fveq1i 2833	. . . 4 ⊢ (1st ‘A) = ({⟨x, y⟩∣y = ∪dom {x}} ‘A)
11	8, 10	syl5eq 1136	. . 3 ⊢ ((A ∈ V ∧ ∪dom {A} ∈ V) → (1st ‘A) = ∪dom {A})
12	4, 11	mpan2 519	. 2 ⊢ (A ∈ V → (1st ‘A) = ∪dom {A})
13		fvprc 2829	. . 3 ⊢ (¬ A ∈ V → (1st ‘A) = ∅)
14		snprc 1838	. . . . . . . 8 ⊢ (¬ A ∈ V ↔ {A} = ∅)
15	14	biimp 133	. . . . . . 7 ⊢ (¬ A ∈ V → {A} = ∅)
16	15	dmeqd 2533	. . . . . 6 ⊢ (¬ A ∈ V → dom {A} = dom ∅)
17		dm0 2542	. . . . . 6 ⊢ dom ∅ = ∅
18	16, 17	syl6eq 1140	. . . . 5 ⊢ (¬ A ∈ V → dom {A} = ∅)
19	18	unieqd 1929	. . . 4 ⊢ (¬ A ∈ V → ∪dom {A} = ∪∅)
20		uni0 1938	. . . 4 ⊢ ∪∅ = ∅
21	19, 20	syl6eq 1140	. . 3 ⊢ (¬ A ∈ V → ∪dom {A} = ∅)
22	13, 21	eqtr4d 1131	. 2 ⊢ (¬ A ∈ V → (1st ‘A) = ∪dom {A})
23	12, 22	pm2.61i 110	1 ⊢ (1st ‘A) = ∪dom {A}

Syntax hints:  ¬ wn 1   ∧ wa 196   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808  ∪cuni 1919  {copab 2055  dom cdm 2410   ‘cfv 2422  1^st c1st 3085

This theorem is referenced by:  op1st 3091  elxp6 3093  1st2val 3097

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-1st 3087

metamath.org