Theorem elrnopab 2884

Description: Membership in the range of an operation abstraction.

Hypotheses

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

Assertion

Ref	Expression
elrnopab	⊢ (C ∈ ran F ↔ ∃x ∈ A C = B)

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

Proof of Theorem elrnopab

Step	Hyp	Ref	Expression
1		elrnopab.1	. . . . 5 ⊢ B ∈ V
2	1	eueq1 1428	. . . 4 ⊢ ∃!y y = B
3	2	a1i 7	. . 3 ⊢ (x ∈ A → ∃!y y = B)
4		elrnopab.2	. . 3 ⊢ F = {⟨x, y⟩∣(x ∈ A ∧ y = B)}
5	3, 4	fnopab 2746	. 2 ⊢ F Fn A
6		fvelrn 2883	. . 3 ⊢ (F Fn A → (C ∈ ran F ↔ ∃z ∈ A (F ‘z) = C))
7		hbopab1 2112	. . . . . . . 8 ⊢ (w ∈ {⟨x, y⟩∣(x ∈ A ∧ y = B)} → ∀x w ∈ {⟨x, y⟩∣(x ∈ A ∧ y = B)})
8	4	eleq2i 1153	. . . . . . . 8 ⊢ (w ∈ F ↔ w ∈ {⟨x, y⟩∣(x ∈ A &and y = B)})
9	8	bial 695	. . . . . . . 8 ⊢ (∀x w ∈ F ↔ ∀x w ∈ {⟨x, y⟩∣(x ∈ A ∧ y = B)})
10	7, 8, 9	3imtr4 192	. . . . . . 7 ⊢ (w ∈ F → ∀x w ∈ F)
11		ax-17 925	. . . . . . 7 ⊢ (w ∈ z → ∀x w ∈ z)
12	10, 11	hbfv 2837	. . . . . 6 ⊢ (w ∈ (F ‘z) → ∀x w ∈ (F ‘z))
13		ax-17 925	. . . . . 6 ⊢ (w ∈ C → ∀x w ∈ C)
14	12, 13	hbeq 1171	. . . . 5 ⊢ ((F ‘z) = C → ∀x(F ‘z) = C)
15		ax-17 925	. . . . 5 ⊢ ((F ‘x) = C → ∀z(F ‘x) = C)
16		fveq2 2832	. . . . . 6 ⊢ (z = x → (F ‘z) = (F ‘x))
17	16	cleq1d 1109	. . . . 5 ⊢ (z = x → ((F ‘z) = C ↔ (F ‘x) = C))
18	14, 15, 17	cbvrex 1332	. . . 4 ⊢ (∃z ∈ A (F ‘z) = C ↔ ∃x ∈ A (F ‘x) = C)
19		fvopab2 2878	. . . . . . . . 9 ⊢ ((x ∈ A ∧ B ∈ V) → ({⟨x, y⟩∣(x ∈ A ∧ y = B)} ‘x) = B)
20	1, 19	mpan2 519	. . . . . . . 8 ⊢ (x ∈ A → ({⟨x, y⟩∣(x ∈ A ∧ y = B)} ‘x) = B)
21	4	fveq1i 2833	. . . . . . . 8 ⊢ (F ‘x) = ({⟨x, y⟩∣(x ∈ A ∧ y = B)} ‘x)
22	20, 21	syl5eq 1136	. . . . . . 7 ⊢ (x ∈ A → (F ‘x) = B)
23	22	cleq1d 1109	. . . . . 6 ⊢ (x ∈ A → ((F ‘x) = C ↔ B = C))
24		cleqcom 1103	. . . . . 6 ⊢ (B = C ↔ C = B)
25	23, 24	syl6bb 414	. . . . 5 ⊢ (x ∈ A → ((F ‘x) = C ↔ C = B))
26	25	birexa 1229	. . . 4 ⊢ (∃x ∈ A (F ‘x) = C ↔ ∃x ∈ A C = B)
27	18, 26	bitr 151	. . 3 ⊢ (∃z ∈ A (F ‘z) = C ↔ ∃x ∈ A C = B)
28	6, 27	syl6bb 414	. 2 ⊢ (F Fn A → (C ∈ ran F ↔ ∃x ∈ A C = B))
29	5, 28	ax-mp 6	1 ⊢ (C ∈ ran F ↔ ∃x ∈ A C = B)

Syntax hints:   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797   ∈ wel 803  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348  {copab 2055  ran crn 2411   Fn wfn 2417   ‘cfv 2422

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-fv 2438

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