Theorem fvelrn 2883

Description: A member of a function's range is a value of the function.

Assertion

Ref	Expression
fvelrn	⊢ (F Fn A → (B ∈ ran F ↔ ∃x ∈ A (F ‘x) = B))

Distinct variable group(s):   x,A   x,F   x,B

Proof of Theorem fvelrn

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . 5 ⊢ (y = B → (y ∈ ran F ↔ B ∈ ran F))
2		cleq2 1110	. . . . . 6 ⊢ (y = B → ((F ‘x) = y ↔ (F ‘x) = B))
3	2	birexdv 1220	. . . . 5 ⊢ (y = B → (∃x ∈ A (F ‘x) = y ↔ ∃x ∈ A (F ‘x) = B))
4	1, 3	bibi12d 477	. . . 4 ⊢ (y = B → ((y ∈ ran F ↔ ∃x ∈ A (F ‘x) = y) ↔ (B ∈ ran F ↔ ∃x ∈ A (F ‘x) = B)))
5	4	imbi2d 464	. . 3 ⊢ (y = B → ((F Fn A → (y ∈ ran F ↔ ∃x ∈ A (F ‘x) = y)) ↔ (F Fn A → (B ∈ ran F ↔ ∃x ∈ A (F ‘x) = B))))
6		visset 1350	. . . . . . . . 9 ⊢ x ∈ V
7		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
8	6, 7	fnbr 2726	. . . . . . . 8 ⊢ ((F Fn A ∧ xFy) → x ∈ A)
9	6	tz6.12-1 2842	. . . . . . . . . . 11 ⊢ ((xFy ∧ ∃!y xFy) → (F ‘x) = y)
10	9	exp 291	. . . . . . . . . 10 ⊢ (xFy → (∃!y xFy → (F ‘x) = y))
11		funeu 2685	. . . . . . . . . . 11 ⊢ ((Fun F ∧ xFy) → ∃!y xFy)
12		fnfun 2721	. . . . . . . . . . 11 ⊢ (F Fn A → Fun F)
13	11, 12	sylan 343	. . . . . . . . . 10 ⊢ ((F Fn A ∧ xFy) → ∃!y xFy)
14	10, 13	syl5 22	. . . . . . . . 9 ⊢ (xFy → ((F Fn A ∧ xFy) → (F ‘x) = y))
15	14	anabsi7 379	. . . . . . . 8 ⊢ ((F Fn A ∧ xFy) → (F ‘x) = y)
16	8, 15	jca 236	. . . . . . 7 ⊢ ((F Fn A ∧ xFy) → (x ∈ A ∧ (F ‘x) = y))
17	16	exp 291	. . . . . 6 ⊢ (F Fn A → (xFy → (x ∈ A ∧ (F ‘x) = y)))
18	7	fnfvbr 2855	. . . . . . . . 9 ⊢ ((F Fn A ∧ x ∈ A) → ((F ‘x) = y ↔ xFy))
19	18	biimpd 135	. . . . . . . 8 ⊢ ((F Fn A ∧ x ∈ A) → ((F ‘x) = y → xFy))
20	19	exp 291	. . . . . . 7 ⊢ (F Fn A → (x ∈ A → ((F ‘x) = y → xFy)))
21	20	imp3a 279	. . . . . 6 ⊢ (F Fn A → ((x ∈ A ∧ (F ‘x) = y) → xFy))
22	17, 21	impbid 397	. . . . 5 ⊢ (F Fn A → (xFy ↔ (x ∈ A ∧ (F ‘x) = y)))
23	22	biexdv 936	. . . 4 ⊢ (F Fn A → (∃x xFy ↔ ∃x(x ∈ A ∧ (F ‘x) = y)))
24	7	elrn2 2563	. . . 4 ⊢ (y ∈ ran F ↔ ∃x xFy)
25		df-rex 1206	. . . 4 ⊢ (∃x ∈ A (F ‘x) = y ↔ ∃x(x ∈ A ∧ (F ‘x) = y))
26	23, 24, 25	3bitr4g 428	. . 3 ⊢ (F Fn A → (y ∈ ran F ↔ ∃x ∈ A (F ‘x) = y))
27	5, 26	vtoclg 1383	. 2 ⊢ (B ∈ V → (F Fn A → (B ∈ ran F ↔ ∃x ∈ A (F ‘x) = B)))
28		elisset 1354	. . . 4 ⊢ (B ∈ ran F → B ∈ V)
29		fvex 2838	. . . . . . 7 ⊢ (F ‘x) ∈ V
30		eleq1 1149	. . . . . . 7 ⊢ ((F ‘x) = B → ((F ‘x) ∈ V ↔ B ∈ V))
31	29, 30	mpbii 168	. . . . . 6 ⊢ ((F ‘x) = B → B ∈ V)
32	31	a1i 7	. . . . 5 ⊢ (x ∈ A → ((F ‘x) = B → B ∈ V))
33	32	r19.23aiv 1284	. . . 4 ⊢ (∃x ∈ A (F ‘x) = B → B ∈ V)
34	28, 33	pm5.21ni 503	. . 3 ⊢ (¬ B ∈ V → (B ∈ ran F ↔ ∃x ∈ A (F ‘x) = B))
35	34	a1d 14	. 2 ⊢ (¬ B ∈ V → (F Fn A → (B ∈ ran F ↔ ∃x ∈ A (F ‘x) = B)))
36	27, 35	pm2.61i 110	1 ⊢ (F Fn A → (B ∈ ran F ↔ ∃x ∈ A (F ‘x) = B))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348   class class class wbr 2054  ran crn 2411  Fun wfun 2416   Fn wfn 2417   ‘cfv 2422

This theorem is referenced by:  elrnopab 2884  chfnrn 2885  ffnfv 2892  fconstfv 2903  isoini 2938  canth 2945  elrnoprab 3054  mapenlem2 3385  inf0 3457  inf3lem6 3469  aceq5 3563  zornlem4 3606  isinfcard 3692  om2uzran 4655  ruclem33 4917  ruclem35 4919  ruclem37 4921

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-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-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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