Theorem f1fvf 2917

Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43.

Hypotheses

Ref	Expression
f1fvf.1	⊢ (z ∈ F → ∀x z ∈ F)
f1fvf.2	⊢ (z ∈ F → ∀y z ∈ F)

Assertion

Ref	Expression
f1fvf	⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))

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

Proof of Theorem f1fvf

Step	Hyp	Ref	Expression
1		f1fv 2916	. 2 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀w ∈ A ∀v ∈ A ((F ‘w) = (F ‘v) → w = v)))
2		f1fvf.2	. . . . . . . . 9 ⊢ (z ∈ F → ∀y z ∈ F)
3		ax-17 925	. . . . . . . . 9 ⊢ (z ∈ w → ∀y z ∈ w)
4	2, 3	hbfv 2837	. . . . . . . 8 ⊢ (z ∈ (F ‘w) → ∀y z ∈ (F ‘w))
5		ax-17 925	. . . . . . . . 9 ⊢ (z ∈ v → ∀y z ∈ v)
6	2, 5	hbfv 2837	. . . . . . . 8 ⊢ (z ∈ (F ‘v) → ∀y z ∈ (F ‘v))
7	4, 6	hbeq 1171	. . . . . . 7 ⊢ ((F ‘w) = (F ‘v) → ∀y(F ‘w) = (F ‘v))
8		ax-17 925	. . . . . . 7 ⊢ (w = v → ∀y w = v)
9	7, 8	hbim 702	. . . . . 6 ⊢ (((F ‘w) = (F ‘v) → w = v) → ∀y((F ‘w) = (F ‘v) → w = v))
10		ax-17 925	. . . . . . 7 ⊢ ((F ‘w) = (F ‘y) → ∀v(F ‘w) = (F ‘y))
11		ax-17 925	. . . . . . 7 ⊢ (w = y → ∀v w = y)
12	10, 11	hbim 702	. . . . . 6 ⊢ (((F ‘w) = (F ‘y) → w = y) → ∀v((F ‘w) = (F ‘y) → w = y))
13		fveq2 2832	. . . . . . . 8 ⊢ (v = y → (F ‘v) = (F ‘y))
14	13	cleq2d 1112	. . . . . . 7 ⊢ (v = y → ((F ‘w) = (F ‘v) ↔ (F ‘w) = (F ‘y)))
15		cleq2 1110	. . . . . . 7 ⊢ (v = y → (w = v ↔ w = y))
16	14, 15	imbi12d 474	. . . . . 6 ⊢ (v = y → (((F ‘w) = (F ‘v) → w = v) ↔ ((F ‘w) = (F ‘y) → w = y)))
17	9, 12, 16	cbvral 1331	. . . . 5 ⊢ (∀v ∈ A ((F ‘w) = (F ‘v) → w = v) ↔ ∀y ∈ A ((F ‘w) = (F ‘y) → w = y))
18	17	biral 1223	. . . 4 ⊢ (∀w ∈ A ∀v ∈ A ((F ‘w) = (F ‘v) → w = v) ↔ ∀w ∈ A ∀y ∈ A ((F ‘w) = (F ‘y) → w = y))
19		ax-17 925	. . . . . 6 ⊢ (y ∈ A → ∀x y ∈ A)
20		f1fvf.1	. . . . . . . . 9 ⊢ (z ∈ F → ∀x z ∈ F)
21		ax-17 925	. . . . . . . . 9 ⊢ (z ∈ w → ∀x z ∈ w)
22	20, 21	hbfv 2837	. . . . . . . 8 ⊢ (z ∈ (F ‘w) → ∀x z ∈ (F ‘w))
23		ax-17 925	. . . . . . . . 9 ⊢ (z ∈ y → ∀x z ∈ y)
24	20, 23	hbfv 2837	. . . . . . . 8 ⊢ (z ∈ (F ‘y) → ∀x z ∈ (F ‘y))
25	22, 24	hbeq 1171	. . . . . . 7 ⊢ ((F ‘w) = (F ‘y) → ∀x(F ‘w) = (F ‘y))
26		ax-17 925	. . . . . . 7 ⊢ (w = y → ∀x w = y)
27	25, 26	hbim 702	. . . . . 6 ⊢ (((F ‘w) = (F ‘y) → w = y) → ∀x((F ‘w) = (F ‘y) → w = y))
28	19, 27	hbral 1236	. . . . 5 ⊢ (∀y ∈ A ((F ‘w) = (F ‘y) → w = y) → ∀x∀y ∈ A ((F ‘w) = (F ‘y) → w = y))
29		ax-17 925	. . . . 5 ⊢ (∀y ∈ A ((F ‘x) = (F ‘y) → x = y) → ∀w∀y ∈ A ((F ‘x) = (F ‘y) → x = y))
30		fveq2 2832	. . . . . . . 8 ⊢ (w = x → (F ‘w) = (F ‘x))
31	30	cleq1d 1109	. . . . . . 7 ⊢ (w = x → ((F ‘w) = (F ‘y) ↔ (F ‘x) = (F ‘y)))
32		cleq1 1107	. . . . . . 7 ⊢ (w = x → (w = y ↔ x = y))
33	31, 32	imbi12d 474	. . . . . 6 ⊢ (w = x → (((F ‘w) = (F ‘y) → w = y) ↔ ((F ‘x) = (F ‘y) → x = y)))
34	33	biraldv 1219	. . . . 5 ⊢ (w = x → (∀y ∈ A ((F ‘w) = (F ‘y) → w = y) ↔ ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
35	28, 29, 34	cbvral 1331	. . . 4 ⊢ (∀w ∈ A ∀y ∈ A ((F ‘w) = (F ‘y) → w = y) ↔ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y))
36	18, 35	bitr 151	. . 3 ⊢ (∀w ∈ A ∀v ∈ A ((F ‘w) = (F ‘v) → w = v) ↔ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y))
37	36	anbi2i 367	. 2 ⊢ ((F:A–→B ∧ ∀w ∈ A ∀v ∈ A ((F ‘w) = (F ‘v) → w = v)) ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
38	1, 37	bitr 151	1 ⊢ (F:A–1-1→B ↔ (F:A–→B ∧ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797   ∈ wel 803   = wceq 1091   ∈ wcel 1092  ∀wral 1201  –→wf 2418  –1-1→wf1 2419   ‘cfv 2422

This theorem is referenced by:  dom2d 3307

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-f 2434  df-f1 2435  df-fv 2438

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