Theorem f1stres 3096

Description: Mapping of a restriction of the 1^st (first member of an ordered pair) function.

Assertion

Ref	Expression
f1stres	⊢ (1st ↾ (A × B)):(A × B)–→A

Proof of Theorem f1stres

Step	Hyp	Ref	Expression
1		visset 1350	. . . . . . . 8 ⊢ y ∈ V
2	1	op1sta 2635	. . . . . . 7 ⊢ ∪dom {⟨y, z⟩} = y
3	2	eleq1i 1152	. . . . . 6 ⊢ (∪dom {⟨y, z⟩} ∈ A ↔ y ∈ A)
4	3	biimpr 134	. . . . 5 ⊢ (y ∈ A → ∪dom {⟨y, z⟩} ∈ A)
5	4	adantr 306	. . . 4 ⊢ ((y ∈ A ∧ z ∈ B) → ∪dom {⟨y, z⟩} ∈ A)
6	5	rgen2a 1264	. . 3 ⊢ ∀y ∈ A ∀z ∈ B ∪dom {⟨y, z⟩} ∈ A
7		sneq 1816	. . . . . . 7 ⊢ (x = ⟨y, z⟩ → {x} = {⟨y, z⟩})
8	7	dmeqd 2533	. . . . . 6 ⊢ (x = ⟨y, z⟩ → dom {x} = dom {⟨y, z⟩})
9	8	unieqd 1929	. . . . 5 ⊢ (x = ⟨y, z⟩ → ∪dom {x} = ∪dom {⟨y, z⟩})
10	9	eleq1d 1155	. . . 4 ⊢ (x = ⟨y, z⟩ → (∪dom {x} ∈ A ↔ ∪dom {⟨y, z⟩} ∈ A))
11	10	ralxp 2456	. . 3 ⊢ (∀x ∈ (A × B)∪dom {x} ∈ A ↔ ∀y ∈ A ∀z ∈ B ∪dom {⟨y, z⟩} ∈ A)
12	6, 11	mpbir 165	. 2 ⊢ ∀x ∈ (A × B)∪dom {x} ∈ A
13		df-1st 3087	. . . . 5 ⊢ 1st = {⟨x, y⟩∣y = ∪dom {x}}
14		reseq1 2575	. . . . 5 ⊢ (1st = {⟨x, y⟩∣y = ∪dom {x}} → (1st ↾ (A × B)) = ({⟨x, y⟩∣y = ∪dom {x}} ↾ (A × B)))
15	13, 14	ax-mp 6	. . . 4 ⊢ (1st ↾ (A × B)) = ({⟨x, y⟩∣y = ∪dom {x}} ↾ (A × B))
16		resopab 2598	. . . 4 ⊢ ({⟨x, y⟩∣y = ∪dom {x}} ↾ (A × B)) = {⟨x, y⟩∣(x ∈ (A × B) ∧ y = ∪dom {x})}
17		cleqid 1102	. . . 4 ⊢ {⟨x, y⟩∣(x ∈ (A × B) ∧ y = ∪dom {x})} = {⟨x, y⟩∣(x ∈ (A × B) ∧ y = ∪dom {x})}
18	15, 16, 17	3eqtr 1123	. . 3 ⊢ (1st ↾ (A × B)) = {⟨x, y⟩∣(x ∈ (A × B) ∧ y = ∪dom {x})}
19	18	fopab2 2891	. 2 ⊢ (∀x ∈ (A × B)∪dom {x} ∈ A ↔ (1st ↾ (A × B)):(A × B)–→A)
20	12, 19	mpbi 164	1 ⊢ (1st ↾ (A × B)):(A × B)–→A

Syntax hints:   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {csn 1808  ⟨cop 1810  ∪cuni 1919  {copab 2055   × cxp 2408  dom cdm 2410   ↾ cres 2412  –→wf 2418  1^st c1st 3085

This theorem is referenced by:  ruclem17 4901

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-ral 1205  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-fn 2433  df-f 2434  df-fv 2438  df-1st 3087

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