Theorem f1fvf 3889

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 groups:   x,y,A   z,F   x,z,y

Proof of Theorem f1fvf

Step	Hyp	Ref	Expression
1		f1fv 3888	. 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-17 975	. . . . . . . . 9 ⊢ (z ∈ w → ∀y z ∈ w)
4	2, 3	hbfv 3743	. . . . . . . 8 ⊢ (z ∈ (F ‘w) → ∀y z ∈ (F ‘w))
5		ax-17 975	. . . . . . . . 9 ⊢ (z ∈ v → ∀y z ∈ v)
6	2, 5	hbfv 3743	. . . . . . . 8 ⊢ (z ∈ (F ‘v) → ∀y z ∈ (F ‘v))
7	4, 6	hbeq 1572	. . . . . . 7 ⊢ ((F ‘w) = (F ‘v) → ∀y(F ‘w) = (F ‘v))
8		ax-17 975	. . . . . . 7 ⊢ (w = v → ∀y w = v)
9	7, 8	hbim 1013	. . . . . 6 ⊢ (((F ‘w) = (F ‘v) → w = v) → ∀y((F ‘w) = (F ‘v) → w = v))
10		ax-17 975	. . . . . . 7 ⊢ ((F ‘w) = (F ‘y) → ∀v(F ‘w) = (F ‘y))
11		ax-17 975	. . . . . . 7 ⊢ (w = y → ∀v w = y)
12	10, 11	hbim 1013	. . . . . 6 ⊢ (((F ‘w) = (F ‘y) → w = y) → ∀v((F ‘w) = (F ‘y) → w = y))
13		fveq2 3738	. . . . . . . 8 ⊢ (v = y → (F ‘v) = (F ‘y))
14	13	eqeq2d 1493	. . . . . . 7 ⊢ (v = y → ((F ‘w) = (F ‘v) ↔ (F ‘w) = (F ‘y)))
15		eqeq2 1491	. . . . . . 7 ⊢ (v = y → (w = v ↔ w = y))
16	14, 15	imbi12d 629	. . . . . 6 ⊢ (v = y → (((F ‘w) = (F ‘v) → w = v) ↔ ((F ‘w) = (F ‘y) → w = y)))
17	9, 12, 16	cbvral 1805	. . . . 5 ⊢ (∀v ∈ A ((F ‘w) = (F ‘v) → w = v) ↔ ∀y ∈ A ((F ‘w) = (F ‘y) → w = y))
18	17	ralbii 1674	. . . 4 ⊢ (∀w ∈ A ∀v ∈ A ((F ‘w) = (F ‘v) → w = v) ↔ ∀w ∈ A ∀y ∈ A ((F ‘w) = (F ‘y) → w = y))
19		ax-17 975	. . . . . 6 ⊢ (y ∈ A → ∀x y ∈ A)
20		f1fvf.1	. . . . . . . . 9 ⊢ (z ∈ F → ∀x z ∈ F)
21		ax-17 975	. . . . . . . . 9 ⊢ (z ∈ w → ∀x z ∈ w)
22	20, 21	hbfv 3743	. . . . . . . 8 ⊢ (z ∈ (F ‘w) → ∀x z ∈ (F ‘w))
23		ax-17 975	. . . . . . . . 9 ⊢ (z ∈ y → ∀x z ∈ y)
24	20, 23	hbfv 3743	. . . . . . . 8 ⊢ (z ∈ (F ‘y) → ∀x z ∈ (F ‘y))
25	22, 24	hbeq 1572	. . . . . . 7 ⊢ ((F ‘w) = (F ‘y) → ∀x(F ‘w) = (F ‘y))
26		ax-17 975	. . . . . . 7 ⊢ (w = y → ∀x w = y)
27	25, 26	hbim 1013	. . . . . 6 ⊢ (((F ‘w) = (F ‘y) → w = y) → ∀x((F ‘w) = (F ‘y) → w = y))
28	19, 27	hbral 1693	. . . . 5 ⊢ (∀y ∈ A ((F ‘w) = (F ‘y) → w = y) → ∀x∀y ∈ A ((F ‘w) = (F ‘y) → w = y))
29		ax-17 975	. . . . 5 ⊢ (∀y ∈ A ((F ‘x) = (F ‘y) → x = y) → ∀w∀y ∈ A ((F ‘x) = (F ‘y) → x = y))
30		fveq2 3738	. . . . . . . 8 ⊢ (w = x → (F ‘w) = (F ‘x))
31	30	eqeq1d 1490	. . . . . . 7 ⊢ (w = x → ((F ‘w) = (F ‘y) ↔ (F ‘x) = (F ‘y)))
32		eqeq1 1488	. . . . . . 7 ⊢ (w = x → (w = y ↔ x = y))
33	31, 32	imbi12d 629	. . . . . 6 ⊢ (w = x → (((F ‘w) = (F ‘y) → w = y) ↔ ((F ‘x) = (F ‘y) → x = y)))
34	33	ralbidv 1670	. . . . 5 ⊢ (w = x → (∀y ∈ A ((F ‘w) = (F ‘y) → w = y) ↔ ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))
35	28, 29, 34	cbvral 1805	. . . 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 173	. . 3 ⊢ (∀w ∈ A ∀v ∈ A ((F ‘w) = (F ‘v) → w = v) ↔ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y))
37	36	anbi2i 483	. 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 173	1 ⊢ (F:A–1-1→B ↔ (F:A–→B ⋀ ∀x ∈ A ∀y ∈ A ((F ‘x) = (F ‘y) → x = y)))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 958   = wceq 960   ∈ wcel 962  ∀wral 1652  –→wf 3192  –1-1→wf1 3193   ‘cfv 3196

This theorem is referenced by:  dom2d 4418

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-9 969  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1129  ax-10o 1146  ax-16 1216  ax-11o 1224  ax-ext 1466  ax-sep 2716  ax-pow 2756  ax-pr 2793  ax-un 2880

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1178  df-eu 1388  df-mo 1389  df-clab 1471  df-cleq 1476  df-clel 1479  df-ne 1594  df-ral 1656  df-rex 1657  df-v 1819  df-dif 2058  df-un 2059  df-in 2060  df-ss 2062  df-nul 2290  df-pw 2412  df-sn 2422  df-pr 2423  df-op 2426  df-uni 2516  df-br 2633  df-opab 2680  df-id 2849  df-xp 3198  df-rel 3199  df-cnv 3200  df-co 3201  df-dm 3202  df-rn 3203  df-res 3204  df-ima 3205  df-fun 3206  df-fn 3207  df-f 3208  df-f1 3209  df-fv 3212

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