Theorem dff13f 5976

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 5974	. 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 2392	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑤
4	2, 3	nffv 5705	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐹‘𝑤)
5		nfcv 2392	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑣
6	2, 5	nffv 5705	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐹‘𝑣)
7	4, 6	nfeq 2400	. . . . . . 7 ⊢ Ⅎ𝑦(𝐹‘𝑤) = (𝐹‘𝑣)
8		nfv 1581	. . . . . . 7 ⊢ Ⅎ𝑦 𝑤 = 𝑣
9	7, 8	nfim 1625	. . . . . 6 ⊢ Ⅎ𝑦((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣)
10		nfv 1581	. . . . . 6 ⊢ Ⅎ𝑣((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
11		fveq2 5695	. . . . . . . 8 ⊢ (𝑣 = 𝑦 → (𝐹‘𝑣) = (𝐹‘𝑦))
12	11	eqeq2d 2250	. . . . . . 7 ⊢ (𝑣 = 𝑦 → ((𝐹‘𝑤) = (𝐹‘𝑣) ↔ (𝐹‘𝑤) = (𝐹‘𝑦)))
13		equequ2 1765	. . . . . . 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 2782	. . . . 5 ⊢ (∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦))
16	15	ralbii 2556	. . . 4 ⊢ (∀𝑤 ∈ 𝐴 ∀𝑣 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑣) → 𝑤 = 𝑣) ↔ ∀𝑤 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦))
17		nfcv 2392	. . . . . 6 ⊢ Ⅎ𝑥𝐴
18		dff13f.1	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐹
19		nfcv 2392	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑤
20	18, 19	nffv 5705	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑤)
21		nfcv 2392	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑦
22	18, 21	nffv 5705	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑦)
23	20, 22	nfeq 2400	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑤) = (𝐹‘𝑦)
24		nfv 1581	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 = 𝑦
25	23, 24	nfim 1625	. . . . . 6 ⊢ Ⅎ𝑥((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
26	17, 25	nfralxy 2588	. . . . 5 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦)
27		nfv 1581	. . . . 5 ⊢ Ⅎ𝑤∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)
28		fveq2 5695	. . . . . . . 8 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
29	28	eqeq1d 2247	. . . . . . 7 ⊢ (𝑤 = 𝑥 → ((𝐹‘𝑤) = (𝐹‘𝑦) ↔ (𝐹‘𝑥) = (𝐹‘𝑦)))
30		equequ1 1764	. . . . . . 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 2550	. . . . 5 ⊢ (𝑤 = 𝑥 → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑤) = (𝐹‘𝑦) → 𝑤 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
33	26, 27, 32	cbvral 2782	. . . 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 461	. 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)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  Ⅎwnfc 2379  ∀wral 2528  ⟶wf 5373  –1-1→wf1 5374  ‘cfv 5377

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fv 5385

This theorem is used by:  f1mpt  5977  dom2lem  7058