Theorem 1st2val 3097

Description: Value of an alternate definition of the 1^st function.

Assertion

Ref	Expression
1st2val	⊢ ({⟨⟨x, y⟩, z⟩∣z = x} ‘A) = (1st ‘A)

Distinct variable group(s):   x,y,z,A

Proof of Theorem 1st2val

Step	Hyp	Ref	Expression
1		df-opr 3003	. . . . . 6 ⊢ (w{⟨⟨x, y⟩, z⟩∣z = x}v) = ({⟨⟨x, y⟩, z⟩∣z = x} ‘⟨w, v⟩)
2		visset 1350	. . . . . . 7 ⊢ w ∈ V
3		visset 1350	. . . . . . 7 ⊢ v ∈ V
4		id 9	. . . . . . . 8 ⊢ (x = w → x = w)
5		cleqid 1102	. . . . . . . . 9 ⊢ w = w
6	5	a1i 7	. . . . . . . 8 ⊢ (y = v → w = w)
7		visset 1350	. . . . . . . . . . 11 ⊢ x ∈ V
8		visset 1350	. . . . . . . . . . 11 ⊢ y ∈ V
9	7, 8	pm3.2i 234	. . . . . . . . . 10 ⊢ (x ∈ V ∧ y ∈ V)
10	9	biantrur 544	. . . . . . . . 9 ⊢ (z = x ↔ ((x ∈ V ∧ y ∈ V) ∧ z = x))
11	10	bioprabi 3027	. . . . . . . 8 ⊢ {⟨⟨x, y⟩, z⟩∣z = x} = {⟨⟨x, y⟩, z⟩∣((x ∈ V ∧ y ∈ V) ∧ z = x)}
12	2, 4, 6, 11	oprabval2 3051	. . . . . . 7 ⊢ ((w ∈ V ∧ v ∈ V) → (w{⟨⟨x, y⟩, z⟩∣z = x}v) = w)
13	2, 3, 12	mp2an 520	. . . . . 6 ⊢ (w{⟨⟨x, y⟩, z⟩∣z = x}v) = w
14	1, 13	eqtr3 1121	. . . . 5 ⊢ ({⟨⟨x, y⟩, z⟩∣z = x} ‘⟨w, v⟩) = w
15	2	op1st 3091	. . . . 5 ⊢ (1st ‘⟨w, v⟩) = w
16	14, 15	eqtr4 1122	. . . 4 ⊢ ({⟨⟨x, y⟩, z⟩∣z = x} ‘⟨w, v⟩) = (1st ‘⟨w, v⟩)
17		fveq2 2832	. . . . 5 ⊢ (⟨w, v⟩ = A → ({⟨⟨x, y⟩, z⟩∣z = x} ‘⟨w, v⟩) = ({⟨⟨x, y⟩, z⟩∣z = x} ‘A))
18		fveq2 2832	. . . . 5 ⊢ (⟨w, v⟩ = A → (1st ‘⟨w, v⟩) = (1st ‘A))
19	17, 18	cleq12d 1115	. . . 4 ⊢ (⟨w, v⟩ = A → (({⟨⟨x, y⟩, z⟩∣z = x} ‘⟨w, v⟩) = (1st ‘⟨w, v⟩) ↔ ({⟨⟨x, y⟩, z⟩∣z = x} ‘A) = (1st ‘A)))
20	16, 19	mpbii 168	. . 3 ⊢ (⟨w, v⟩ = A → ({⟨⟨x, y⟩, z⟩∣z = x} ‘A) = (1st ‘A))
21	20	19.23aivv 953	. 2 ⊢ (∃w∃v⟨w, v⟩ = A → ({⟨⟨x, y⟩, z⟩∣z = x} ‘A) = (1st ‘A))
22		a9e 809	. . . . . . . . . 10 ⊢ ∃z z = x
23	9, 22	2th 540	. . . . . . . . 9 ⊢ ((x ∈ V ∧ y ∈ V) ↔ ∃z z = x)
24	23	biopabi 2103	. . . . . . . 8 ⊢ {⟨x, y⟩∣(x ∈ V ∧ y ∈ V)} = {⟨x, y⟩∣∃z z = x}
25		df-xp 2424	. . . . . . . 8 ⊢ (V × V) = {⟨x, y⟩∣(x ∈ V ∧ y ∈ V)}
26		dmoprab 3031	. . . . . . . 8 ⊢ dom {⟨⟨x, y⟩, z⟩∣z = x} = {⟨x, y⟩∣∃z z = x}
27	24, 25, 26	3eqtr4r 1127	. . . . . . 7 ⊢ dom {⟨⟨x, y⟩, z⟩∣z = x} = (V × V)
28	27	eleq2i 1153	. . . . . 6 ⊢ (A ∈ dom {⟨⟨x, y⟩, z⟩∣z = x} ↔ A ∈ (V × V))
29		elvv 2464	. . . . . 6 ⊢ (A ∈ (V × V) ↔ ∃w∃v A = ⟨w, v⟩)
30		cleqcom 1103	. . . . . . 7 ⊢ (A = ⟨w, v⟩ ↔ ⟨w, v⟩ = A)
31	30	bi2ex 734	. . . . . 6 ⊢ (∃w∃v A = ⟨w, v⟩ ↔ ∃w∃v⟨w, v⟩ = A)
32	28, 29, 31	3bitr 155	. . . . 5 ⊢ (A ∈ dom {⟨⟨x, y⟩, z⟩∣z = x} ↔ ∃w∃v⟨w, v⟩ = A)
33	32	negbii 162	. . . 4 ⊢ (¬ A ∈ dom {⟨⟨x, y⟩, z⟩∣z = x} ↔ ¬ ∃w∃v⟨w, v⟩ = A)
34		ndmfv 2848	. . . 4 ⊢ (¬ A ∈ dom {⟨⟨x, y⟩, z⟩∣z = x} → ({⟨⟨x, y⟩, z⟩∣z = x} ‘A) = ∅)
35	33, 34	sylbir 176	. . 3 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → ({⟨⟨x, y⟩, z⟩∣z = x} ‘A) = ∅)
36		n0 1714	. . . . . . . . 9 ⊢ (¬ dom {A} = ∅ ↔ ∃w w ∈ dom {A})
37	2	eldm2 2528	. . . . . . . . . . 11 ⊢ (w ∈ dom {A} ↔ ∃v⟨w, v⟩ ∈ {A})
38		opex 1893	. . . . . . . . . . . . 13 ⊢ ⟨w, v⟩ ∈ V
39	38	elsnc 1826	. . . . . . . . . . . 12 ⊢ (⟨w, v⟩ ∈ {A} ↔ ⟨w, v⟩ = A)
40	39	biex 733	. . . . . . . . . . 11 ⊢ (∃v⟨w, v⟩ ∈ {A} ↔ ∃v⟨w, v⟩ = A)
41	37, 40	bitr 151	. . . . . . . . . 10 ⊢ (w ∈ dom {A} ↔ ∃v⟨w, v⟩ = A)
42	41	biex 733	. . . . . . . . 9 ⊢ (∃w w ∈ dom {A} ↔ ∃w∃v⟨w, v⟩ = A)
43	36, 42	bitr 151	. . . . . . . 8 ⊢ (¬ dom {A} = ∅ ↔ ∃w∃v⟨w, v⟩ = A)
44	43	biimp 133	. . . . . . 7 ⊢ (¬ dom {A} = ∅ → ∃w∃v⟨w, v⟩ = A)
45	44	con1i 88	. . . . . 6 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → dom {A} = ∅)
46	45	unieqd 1929	. . . . 5 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → ∪dom {A} = ∪∅)
47		uni0 1938	. . . . 5 ⊢ ∪∅ = ∅
48	46, 47	syl6eq 1140	. . . 4 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → ∪dom {A} = ∅)
49		1stval 3089	. . . 4 ⊢ (1st ‘A) = ∪dom {A}
50	48, 49	syl5eq 1136	. . 3 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → (1st ‘A) = ∅)
51	35, 50	eqtr4d 1131	. 2 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → ({⟨⟨x, y⟩, z⟩∣z = x} ‘A) = (1st ‘A))
52	21, 51	pm2.61i 110	1 ⊢ ({⟨⟨x, y⟩, z⟩∣z = x} ‘A) = (1st ‘A)

Syntax hints:  ¬ wn 1   ∧ wa 196  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ∅c0 1707  {csn 1808  ⟨cop 1810  ∪cuni 1919  {copab 2055   × cxp 2408  dom cdm 2410   ‘cfv 2422  (class class class)co 3001  {copab2 3002  1^st c1st 3085

This theorem is referenced by:  df1st2 3098

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-3an 583  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-opr 3003  df-oprab 3004  df-1st 3087

metamath.org