Theorem dff13f 5820

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 5818	. 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 2339	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑤
4	2, 3	nffv 5571	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐹‘𝑤)
5		nfcv 2339	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑣
6	2, 5	nffv 5571	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐹‘𝑣)
7	4, 6	nfeq 2347	. . . . . . 7 ⊢ Ⅎ𝑦(𝐹‘𝑤) = (𝐹‘𝑣)
8		nfv 1542	. . . . . . 7 ⊢ Ⅎ𝑦 𝑤 = 𝑣
9	7, 8	nfim 1586	. . . . . 6 ⊢ Ⅎ𝑦((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣)
10		nfv 1542	. . . . . 6 ⊢ Ⅎ𝑣((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
11		fveq2 5561	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
12	11	eqeq2d 2208	. . . . . . 7 ⊢ (𝑣 = 𝑦 → ((𝐹‘𝑤) = (𝐹‘𝑣) ↔ (𝐹‘𝑤) = (𝐹‘𝑦)))
13		equequ2 1727	. . . . . . 7 ⊢ (𝑣 = 𝑦 → (𝑤 = 𝑣 ↔ 𝑤 = 𝑦))
14	12, 13	imbi12d 234	. . . . . 6 ⊢ (𝑣 = 𝑦 → (((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)))
15	9, 10, 14	cbvral 2725	. . . . 5 ⊢ (∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦))
16	15	ralbii 2503	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦))
17		nfcv 2339	. . . . . 6 ⊢ Ⅎ𝑥𝐴
18		dff13f.1	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
19		nfcv 2339	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑤
20	18, 19	nffv 5571	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑤)
21		nfcv 2339	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
22	18, 21	nffv 5571	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
23	20, 22	nfeq 2347	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑤) = (𝐹‘𝑦)
24		nfv 1542	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 = 𝑦
25	23, 24	nfim 1586	. . . . . 6 ⊢ Ⅎ𝑥((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
26	17, 25	nfralxy 2535	. . . . 5 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
27		nfv 1542	. . . . 5 ⊢ Ⅎ𝑤∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
28		fveq2 5561	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
29	28	eqeq1d 2205	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑤) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
30		equequ1 1726	. . . . . . 7 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 ↔ 𝑥 = 𝑦))
31	29, 30	imbi12d 234	. . . . . 6 ⊢ (𝑤 = 𝑥 → (((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
32	31	ralbidv 2497	. . . . 5 ⊢ (𝑤 = 𝑥 → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
33	26, 27, 32	cbvral 2725	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
34	16, 33	bitri 184	. . 3 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
35	34	anbi2i 457	. 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)))
36	1, 35	bitri 184	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364  Ⅎwnfc 2326  ∀wral 2475  ⟶wf 5255  –1-1→wf1 5256  ‘cfv 5259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fv 5267

This theorem is referenced by:  f1mpt  5821  dom2lem  6840