Theorem dff13f 7257

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 7256	. 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 2923	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑤
4	2, 3	nffv 6893	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐹‘𝑤)
5		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑣
6	2, 5	nffv 6893	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐹‘𝑣)
7	4, 6	nfeq 2936	. . . . . . 7 ⊢ Ⅎ𝑦(𝐹‘𝑤) = (𝐹‘𝑣)
8		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑦 𝑤 = 𝑣
9	7, 8	nfim 1929	. . . . . 6 ⊢ Ⅎ𝑦((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣)
10		nfv 1947	. . . . . 6 ⊢ Ⅎ𝑣((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
11		fveq2 6883	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
12	11	eqeq2d 2772	. . . . . . 7 ⊢ (𝑣 = 𝑦 → ((𝐹‘𝑤) = (𝐹‘𝑣) ↔ (𝐹‘𝑤) = (𝐹‘𝑦)))
13		equequ2 2059	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (𝑤 = 𝑣 ↔ 𝑤 = 𝑦))
14	12, 13	imbi12d 347	. . . . . 6 ⊢ (𝑣 = 𝑦 → (((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)))
15	9, 10, 14	cbvralw 3305	. . . . 5 ⊢ (∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦))
16	15	ralbii 3109	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦))
17		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥𝐴
18		dff13f.1	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
19		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑤
20	18, 19	nffv 6893	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑤)
21		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
22	18, 21	nffv 6893	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
23	20, 22	nfeq 2936	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑤) = (𝐹‘𝑦)
24		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 = 𝑦
25	23, 24	nfim 1929	. . . . . 6 ⊢ Ⅎ𝑥((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
26	17, 25	nfralw 3310	. . . . 5 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
27		nfv 1947	. . . . 5 ⊢ Ⅎ𝑤∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
28		fveqeq2 6892	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑤) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
29		equequ1 2058	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 ↔ 𝑥 = 𝑦))
30	28, 29	imbi12d 347	. . . . . 6 ⊢ (𝑤 = 𝑥 → (((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
31	30	ralbidv 3186	. . . . 5 ⊢ (𝑤 = 𝑥 → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
32	26, 27, 31	cbvralw 3305	. . . 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 278	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
34	33	anbi2i 635	. 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 278	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  Ⅎwnfc 2908  ∀wral 3077  ⟶wf 6533  –1-1→wf1 6534  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fv 6545

This theorem is used by:  f1mpt  7263  dom2lem  9012