Theorem elrnoprab 3054

Description: Membership in the range of an operation abstraction.

Hypotheses

Ref	Expression
elrnoprab.1	⊢ C ∈ V
elrnoprab.2	⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)}

Assertion

Ref	Expression
elrnoprab	⊢ (D ∈ ran F ↔ ∃x ∈ A ∃y ∈ B D = C)

Distinct variable group(s):   x,y,z,A   x,B,y,z   z,C   x,D,y

Proof of Theorem elrnoprab

Step	Hyp	Ref	Expression
1		elrnoprab.1	. . 3 ⊢ C ∈ V
2		elrnoprab.2	. . 3 ⊢ F = {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)}
3	1, 2	fnoprab2 3039	. 2 ⊢ F Fn (A × B)
4		fvelrn 2883	. . 3 ⊢ (F Fn (A × B) → (D ∈ ran F ↔ ∃w ∈ (A × B)(F ‘w) = D))
5		fveq2 2832	. . . . . . . 8 ⊢ (w = ⟨v, u⟩ → (F ‘w) = (F ‘⟨v, u⟩))
6	5	cleq1d 1109	. . . . . . 7 ⊢ (w = ⟨v, u⟩ → ((F ‘w) = D ↔ (F ‘⟨v, u⟩) = D))
7	6	cbvop 2473	. . . . . 6 ⊢ (∃w ∈ (A × B)(F ‘w) = D ↔ ∃v ∈ A ∃u ∈ B (F ‘⟨v, u⟩) = D)
8		ax-17 925	. . . . . . . 8 ⊢ (u ∈ B → ∀x u ∈ B)
9		hboprab1 3023	. . . . . . . . . . 11 ⊢ (w ∈ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} → ∀x w ∈ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)})
10	2	eleq2i 1153	. . . . . . . . . . 11 ⊢ (w ∈ F ↔ w ∈ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)})
11	10	bial 695	. . . . . . . . . . 11 ⊢ (∀x w ∈ F ↔ ∀x w ∈ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)})
12	9, 10, 11	3imtr4 192	. . . . . . . . . 10 ⊢ (w ∈ F → ∀x w ∈ F)
13		ax-17 925	. . . . . . . . . 10 ⊢ (w ∈ ⟨v, u⟩ → ∀x w ∈ ⟨v, u⟩)
14	12, 13	hbfv 2837	. . . . . . . . 9 ⊢ (w ∈ (F ‘⟨v, u⟩) → ∀x w ∈ (F ‘⟨v, u⟩))
15		ax-17 925	. . . . . . . . 9 ⊢ (w ∈ D → ∀x w ∈ D)
16	14, 15	hbeq 1171	. . . . . . . 8 ⊢ ((F ‘⟨v, u⟩) = D → ∀x(F ‘⟨v, u⟩) = D)
17	8, 16	hbrex 1238	. . . . . . 7 ⊢ (∃u ∈ B (F ‘⟨v, u⟩) = D → ∀x∃u ∈ B (F ‘⟨v, u⟩) = D)
18		ax-17 925	. . . . . . 7 ⊢ (∃u ∈ B (F ‘⟨x, u⟩) = D → ∀v∃u ∈ B (F ‘⟨x, u⟩) = D)
19		opeq1 1876	. . . . . . . . . 10 ⊢ (v = x → ⟨v, u⟩ = ⟨x, u⟩)
20	19	fveq2d 2836	. . . . . . . . 9 ⊢ (v = x → (F ‘⟨v, u⟩) = (F ‘⟨x, u⟩))
21	20	cleq1d 1109	. . . . . . . 8 ⊢ (v = x → ((F ‘⟨v, u⟩) = D ↔ (F ‘⟨x, u⟩) = D))
22	21	birexdv 1220	. . . . . . 7 ⊢ (v = x → (∃u ∈ B (F ‘⟨v, u⟩) = D ↔ ∃u ∈ B (F ‘⟨x, u⟩) = D))
23	17, 18, 22	cbvrex 1332	. . . . . 6 ⊢ (∃v ∈ A ∃u ∈ B (F ‘⟨v, u⟩) = D ↔ ∃x ∈ A ∃u ∈ B (F ‘⟨x, u⟩) = D)
24		hboprab2 3024	. . . . . . . . . . 11 ⊢ (w ∈ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)} → ∀y w ∈ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)})
25	10	bial 695	. . . . . . . . . . 11 ⊢ (∀y w ∈ F ↔ ∀y w ∈ {⟨⟨x, y⟩, z⟩∣((x ∈ A ∧ y ∈ B) ∧ z = C)})
26	24, 10, 25	3imtr4 192	. . . . . . . . . 10 ⊢ (w ∈ F → ∀y w ∈ F)
27		ax-17 925	. . . . . . . . . 10 ⊢ (w ∈ ⟨x, u⟩ → ∀y w ∈ ⟨x, u⟩)
28	26, 27	hbfv 2837	. . . . . . . . 9 ⊢ (w ∈ (F ‘⟨x, u⟩) → ∀y w ∈ (F ‘⟨x, u⟩))
29		ax-17 925	. . . . . . . . 9 ⊢ (w ∈ D → ∀y w ∈ D)
30	28, 29	hbeq 1171	. . . . . . . 8 ⊢ ((F ‘⟨x, u⟩) = D → ∀y(F ‘⟨x, u⟩) = D)
31		ax-17 925	. . . . . . . 8 ⊢ ((F ‘⟨x, y⟩) = D → ∀u(F ‘⟨x, y⟩) = D)
32		opeq2 1877	. . . . . . . . . 10 ⊢ (u = y → ⟨x, u⟩ = ⟨x, y⟩)
33	32	fveq2d 2836	. . . . . . . . 9 ⊢ (u = y → (F ‘⟨x, u⟩) = (F ‘⟨x, y⟩))
34	33	cleq1d 1109	. . . . . . . 8 ⊢ (u = y → ((F ‘⟨x, u⟩) = D ↔ (F ‘⟨x, y⟩) = D))
35	30, 31, 34	cbvrex 1332	. . . . . . 7 ⊢ (∃u ∈ B (F ‘⟨x, u⟩) = D ↔ ∃y ∈ B (F ‘⟨x, y⟩) = D)
36	35	birex 1224	. . . . . 6 ⊢ (∃x ∈ A ∃u ∈ B (F ‘⟨x, u⟩) = D ↔ ∃x ∈ A ∃y ∈ B (F ‘⟨x, y⟩) = D)
37	7, 23, 36	3bitr 155	. . . . 5 ⊢ (∃w ∈ (A × B)(F ‘w) = D ↔ ∃x ∈ A ∃y ∈ B (F ‘⟨x, y⟩) = D)
38		df-opr 3003	. . . . . . 7 ⊢ (xFy) = (F ‘⟨x, y⟩)
39	38	cleq1i 1108	. . . . . 6 ⊢ ((xFy) = D ↔ (F ‘⟨x, y⟩) = D)
40	39	bi2rex 1226	. . . . 5 ⊢ (∃x ∈ A ∃y ∈ B (xFy) = D ↔ ∃x ∈ A ∃y ∈ B (F ‘⟨x, y⟩) = D)
41	37, 40	bitr4 154	. . . 4 ⊢ (∃w ∈ (A × B)(F ‘w) = D ↔ ∃x ∈ A ∃y ∈ B (xFy) = D)
42	2	oprabval4g 3053	. . . . . . . 8 ⊢ ((x ∈ A ∧ y ∈ B ∧ C ∈ V) → (xFy) = C)
43	1, 42	mp3an3 641	. . . . . . 7 ⊢ ((x ∈ A ∧ y ∈ B) → (xFy) = C)
44	43	cleq1d 1109	. . . . . 6 ⊢ ((x ∈ A ∧ y ∈ B) → ((xFy) = D ↔ C = D))
45		cleqcom 1103	. . . . . 6 ⊢ (C = D ↔ D = C)
46	44, 45	syl6bb 414	. . . . 5 ⊢ ((x ∈ A ∧ y ∈ B) → ((xFy) = D ↔ D = C))
47	46	bi2rexa 1230	. . . 4 ⊢ (∃x ∈ A ∃y ∈ B (xFy) = D ↔ ∃x ∈ A ∃y ∈ B D = C)
48	41, 47	bitr 151	. . 3 ⊢ (∃w ∈ (A × B)(F ‘w) = D ↔ ∃x ∈ A ∃y ∈ B D = C)
49	4, 48	syl6bb 414	. 2 ⊢ (F Fn (A × B) → (D ∈ ran F ↔ ∃x ∈ A ∃y ∈ B D = C))
50	3, 49	ax-mp 6	1 ⊢ (D ∈ ran F ↔ ∃x ∈ A ∃y ∈ B D = C)

Syntax hints:   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  ⟨cop 1810   × cxp 2408  ran crn 2411   Fn wfn 2417   ‘cfv 2422  (class class class)co 3001  {copab2 3002

This theorem is referenced by:  unxpdomlem 3649  qnnen 4931

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-fn 2433  df-fv 2438  df-opr 3003  df-oprab 3004

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