Theorem dff13f 7129

Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 31-Jul-2003.)

Hypotheses

Ref	Expression
dff13f.1	⊢ Ⅎ𝑥𝐹
dff13f.2	⊢ Ⅎ𝑦𝐹

Assertion

Ref	Expression
dff13f	⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem dff13f

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dff13 7128	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣)))
2		dff13f.2	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐹
3		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑤
4	2, 3	nffv 6784	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐹‘𝑤)
5		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑣
6	2, 5	nffv 6784	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐹‘𝑣)
7	4, 6	nfeq 2920	. . . . . . 7 ⊢ Ⅎ𝑦(𝐹‘𝑤) = (𝐹‘𝑣)
8		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑦 𝑤 = 𝑣
9	7, 8	nfim 1899	. . . . . 6 ⊢ Ⅎ𝑦((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣)
10		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑣((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
11		fveq2 6774	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
12	11	eqeq2d 2749	. . . . . . 7 ⊢ (𝑣 = 𝑦 → ((𝐹‘𝑤) = (𝐹‘𝑣) ↔ (𝐹‘𝑤) = (𝐹‘𝑦)))
13		equequ2 2029	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (𝑤 = 𝑣 ↔ 𝑤 = 𝑦))
14	12, 13	imbi12d 345	. . . . . 6 ⊢ (𝑣 = 𝑦 → (((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)))
15	9, 10, 14	cbvralw 3373	. . . . 5 ⊢ (∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦))
16	15	ralbii 3092	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦))
17		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥𝐴
18		dff13f.1	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
19		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑤
20	18, 19	nffv 6784	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑤)
21		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
22	18, 21	nffv 6784	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
23	20, 22	nfeq 2920	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑤) = (𝐹‘𝑦)
24		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 = 𝑦
25	23, 24	nfim 1899	. . . . . 6 ⊢ Ⅎ𝑥((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
26	17, 25	nfralw 3151	. . . . 5 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
27		nfv 1917	. . . . 5 ⊢ Ⅎ𝑤∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
28		fveqeq2 6783	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑤) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
29		equequ1 2028	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 ↔ 𝑥 = 𝑦))
30	28, 29	imbi12d 345	. . . . . 6 ⊢ (𝑤 = 𝑥 → (((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
31	30	ralbidv 3112	. . . . 5 ⊢ (𝑤 = 𝑥 → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
32	26, 27, 31	cbvralw 3373	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
33	16, 32	bitri 274	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
34	33	anbi2i 623	. 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)))
35	1, 34	bitri 274	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  Ⅎwnfc 2887  ∀wral 3064  ⟶wf 6429  –1-1→wf1 6430  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fv 6441

This theorem is referenced by:  f1mpt  7134  dom2lem  8780