Theorem 2nd2val 4086

Description: Value of an alternate definition of the 2^nd function.

Assertion

Ref	Expression
2nd2val	⊢ ({⟨⟨x, y⟩, z⟩∣z = y} ‘A) = (2nd ‘A)

Distinct variable group:   x,y,z,A

Proof of Theorem 2nd2val

Step	Hyp	Ref	Expression
1		visset 1809	. . . . . 6 ⊢ w ∈ V
2		visset 1809	. . . . . 6 ⊢ v ∈ V
3	1, 2	op2nd 4076	. . . . 5 ⊢ (2nd ‘⟨w, v⟩) = v
4		equid 1124	. . . . . . . 8 ⊢ y = y
5	4	a1i 8	. . . . . . 7 ⊢ (x = w → y = y)
6		id 59	. . . . . . 7 ⊢ (y = v → y = v)
7		eqid 1473	. . . . . . 7 ⊢ {⟨⟨x, y⟩, z⟩∣z = y} = {⟨⟨x, y⟩, z⟩∣z = y}
8	2, 5, 6, 7	oprabval5 4020	. . . . . 6 ⊢ ((w ∈ V ⋀ v ∈ V) → (w{⟨⟨x, y⟩, z⟩∣z = y}v) = v)
9	1, 2, 8	mp2an 696	. . . . 5 ⊢ (w{⟨⟨x, y⟩, z⟩∣z = y}v) = v
10		df-opr 3956	. . . . 5 ⊢ (w{⟨⟨x, y⟩, z⟩∣z = y}v) = ({⟨⟨x, y⟩, z⟩∣z = y} ‘⟨w, v⟩)
11	3, 9, 10	3eqtr2r 1499	. . . 4 ⊢ ({⟨⟨x, y⟩, z⟩∣z = y} ‘⟨w, v⟩) = (2nd ‘⟨w, v⟩)
12		fveq2 3715	. . . . 5 ⊢ (⟨w, v⟩ = A → ({⟨⟨x, y⟩, z⟩∣z = y} ‘⟨w, v⟩) = ({⟨⟨x, y⟩, z⟩∣z = y} ‘A))
13		fveq2 3715	. . . . 5 ⊢ (⟨w, v⟩ = A → (2nd ‘⟨w, v⟩) = (2nd ‘A))
14	12, 13	eqeq12d 1486	. . . 4 ⊢ (⟨w, v⟩ = A → (({⟨⟨x, y⟩, z⟩∣z = y} ‘⟨w, v⟩) = (2nd ‘⟨w, v⟩) ↔ ({⟨⟨x, y⟩, z⟩∣z = y} ‘A) = (2nd ‘A)))
15	11, 14	mpbii 193	. . 3 ⊢ (⟨w, v⟩ = A → ({⟨⟨x, y⟩, z⟩∣z = y} ‘A) = (2nd ‘A))
16	15	19.23aivv 1294	. 2 ⊢ (∃w∃v⟨w, v⟩ = A → ({⟨⟨x, y⟩, z⟩∣z = y} ‘A) = (2nd ‘A))
17		visset 1809	. . . . . . . . . . 11 ⊢ x ∈ V
18		visset 1809	. . . . . . . . . . 11 ⊢ y ∈ V
19	17, 18	pm3.2i 285	. . . . . . . . . 10 ⊢ (x ∈ V ⋀ y ∈ V)
20		a9e 1123	. . . . . . . . . 10 ⊢ ∃z z = y
21	19, 20	2th 717	. . . . . . . . 9 ⊢ ((x ∈ V ⋀ y ∈ V) ↔ ∃z z = y)
22	21	opabbii 2666	. . . . . . . 8 ⊢ {⟨x, y⟩∣(x ∈ V ⋀ y ∈ V)} = {⟨x, y⟩∣∃z z = y}
23		df-xp 3179	. . . . . . . 8 ⊢ (V × V) = {⟨x, y⟩∣(x ∈ V ⋀ y ∈ V)}
24		dmoprab 3993	. . . . . . . 8 ⊢ dom {⟨⟨x, y⟩, z⟩∣z = y} = {⟨x, y⟩∣∃z z = y}
25	22, 23, 24	3eqtr4r 1503	. . . . . . 7 ⊢ dom {⟨⟨x, y⟩, z⟩∣z = y} = (V × V)
26	25	eleq2i 1535	. . . . . 6 ⊢ (A ∈ dom {⟨⟨x, y⟩, z⟩∣z = y} ↔ A ∈ (V × V))
27		elvv 3223	. . . . . 6 ⊢ (A ∈ (V × V) ↔ ∃w∃v A = ⟨w, v⟩)
28		eqcom 1474	. . . . . . 7 ⊢ (A = ⟨w, v⟩ ↔ ⟨w, v⟩ = A)
29	28	2exbii 1050	. . . . . 6 ⊢ (∃w∃v A = ⟨w, v⟩ ↔ ∃w∃v⟨w, v⟩ = A)
30	26, 27, 29	3bitr 177	. . . . 5 ⊢ (A ∈ dom {⟨⟨x, y⟩, z⟩∣z = y} ↔ ∃w∃v⟨w, v⟩ = A)
31	30	negbii 187	. . . 4 ⊢ (¬ A ∈ dom {⟨⟨x, y⟩, z⟩∣z = y} ↔ ¬ ∃w∃v⟨w, v⟩ = A)
32		ndmfv 3736	. . . 4 ⊢ (¬ A ∈ dom {⟨⟨x, y⟩, z⟩∣z = y} → ({⟨⟨x, y⟩, z⟩∣z = y} ‘A) = ∅)
33	31, 32	sylbir 201	. . 3 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → ({⟨⟨x, y⟩, z⟩∣z = y} ‘A) = ∅)
34		n0 2285	. . . . . . . . 9 ⊢ (¬ ran { A} = ∅ ↔ ∃v v ∈ ran { A})
35	2	elrn2 3343	. . . . . . . . . . 11 ⊢ (v ∈ ran { A} ↔ ∃w⟨w, v⟩ ∈ {A})
36		opex 2777	. . . . . . . . . . . . 13 ⊢ ⟨w, v⟩ ∈ V
37	36	elsnc 2427	. . . . . . . . . . . 12 ⊢ (⟨w, v⟩ ∈ {A} ↔ ⟨w, v⟩ = A)
38	37	exbii 1049	. . . . . . . . . . 11 ⊢ (∃w⟨w, v⟩ ∈ {A} ↔ ∃w⟨w, v⟩ = A)
39	35, 38	bitr 173	. . . . . . . . . 10 ⊢ (v ∈ ran { A} ↔ ∃w⟨w, v⟩ = A)
40	39	exbii 1049	. . . . . . . . 9 ⊢ (∃v v ∈ ran { A} ↔ ∃v∃w⟨w, v⟩ = A)
41		excom 1044	. . . . . . . . 9 ⊢ (∃v∃w⟨w, v⟩ = A ↔ ∃w∃v⟨w, v⟩ = A)
42	34, 40, 41	3bitr 177	. . . . . . . 8 ⊢ (¬ ran { A} = ∅ ↔ ∃w∃v⟨w, v⟩ = A)
43	42	biimp 151	. . . . . . 7 ⊢ (¬ ran { A} = ∅ → ∃w∃v⟨w, v⟩ = A)
44	43	con1i 96	. . . . . 6 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → ran { A} = ∅)
45	44	unieqd 2507	. . . . 5 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → ∪ran { A} = ∪∅)
46		uni0 2520	. . . . 5 ⊢ ∪∅ = ∅
47	45, 46	syl6eq 1520	. . . 4 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → ∪ran { A} = ∅)
48		2ndval 4072	. . . 4 ⊢ (2nd ‘A) = ∪ran { A}
49	47, 48	syl5eq 1516	. . 3 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → (2nd ‘A) = ∅)
50	33, 49	eqtr4d 1507	. 2 ⊢ (¬ ∃w∃v⟨w, v⟩ = A → ({⟨⟨x, y⟩, z⟩∣z = y} ‘A) = (2nd ‘A))
51	16, 50	pm2.61i 126	1 ⊢ ({⟨⟨x, y⟩, z⟩∣z = y} ‘A) = (2nd ‘A)

Syntax hints:  ¬ wn 2   ⋀ wa 223   = wceq 954   ∈ wcel 956  ∃wex 978  Vcvv 1807  ∅c0 2276  {csn 2405  ⟨cop 2407  ∪cuni 2498  {copab 2661   × cxp 3163  dom cdm 3165  ran crn 3166   ‘cfv 3177  (class class class)co 3954  {copab2 3955  2^nd c2nd 4068

This theorem is referenced by:  df2nd2 4117

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fv 3193  df-opr 3956  df-oprab 3957  df-2nd 4070

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