Theorem dff13 5428

Description: A one-to-one function in terms of function values. Compare Theorem 4.8(iv) of [Monk1] p. 43. (Contributed by NM, 29-Oct-1996.)

Assertion

Ref	Expression
dff13	⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem dff13

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dff12 5111	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧))
2		ffn 5066	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		vex 2604	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
4		vex 2604	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
5	3, 4	breldm 4557	. . . . . . . . . . . . . 14 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
6		fndm 5018	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
7	6	eleq2d 2148	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
8	5, 7	syl5ib 152	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑧 → 𝑥 ∈ 𝐴))
9		vex 2604	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
10	9, 4	breldm 4557	. . . . . . . . . . . . . 14 ⊢ (𝑦𝐹𝑧 → 𝑦 ∈ dom 𝐹)
11	6	eleq2d 2148	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
12	10, 11	syl5ib 152	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑦𝐹𝑧 → 𝑦 ∈ 𝐴))
13	8, 12	anim12d 328	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
14	13	pm4.71rd 386	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧))))
15		eqcom 2083	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑧)
16		fnbrfvb 5235	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑧 ↔ 𝑥𝐹𝑧))
17	15, 16	syl5bb 190	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑧 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑧))
18		eqcom 2083	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = 𝑧)
19		fnbrfvb 5235	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) = 𝑧 ↔ 𝑦𝐹𝑧))
20	18, 19	syl5bb 190	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧 = (𝐹‘𝑦) ↔ 𝑦𝐹𝑧))
21	17, 20	bi2anan9 570	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) ↔ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧)))
22	21	anandis 556	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) ↔ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧)))
23	22	pm5.32da 439	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧))))
24	14, 23	bitr4d 189	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)))))
25	24	imbi1d 229	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))) → 𝑥 = 𝑦)))
26		impexp 259	. . . . . . . . 9 ⊢ ((((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))) → 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦)))
27	25, 26	syl6bb 194	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦))))
28	27	albidv 1745	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦))))
29		19.21v 1794	. . . . . . . 8 ⊢ (∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦)))
30		funfvex 5212	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
31	30	funfni 5019	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
32		eqvincg 2719	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) ∈ V → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))))
33	31, 32	syl 14	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))))
34	33	imbi1d 229	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦)))
35		19.23v 1804	. . . . . . . . . . 11 ⊢ (∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦))
36	34, 35	syl6rbbr 197	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
37	36	adantrr 462	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
38	37	pm5.74da 431	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
39	29, 38	syl5bb 190	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
40	28, 39	bitrd 186	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
41	40	2albidv 1788	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
42		breq1 3788	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑦𝐹𝑧))
43	42	mo4 2002	. . . . . . 7 ⊢ (∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
44	43	albii 1399	. . . . . 6 ⊢ (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑧∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
45		alrot3 1414	. . . . . 6 ⊢ (∀𝑧∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
46	44, 45	bitri 182	. . . . 5 ⊢ (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
47		r2al 2385	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
48	41, 46, 47	3bitr4g 221	. . . 4 ⊢ (𝐹 Fn 𝐴 → (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
49	2, 48	syl 14	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
50	49	pm5.32i 441	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
51	1, 50	bitri 182	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282   = wceq 1284  ∃wex 1421   ∈ wcel 1433  ∃*wmo 1942  ∀wral 2348  Vcvv 2601   class class class wbr 3785  dom cdm 4363   Fn wfn 4917  ⟶wf 4918  –1-1→wf1 4919  ‘cfv 4922

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fv 4930

This theorem is referenced by:  f1veqaeq  5429  dff13f  5430  dff1o6  5436  fcof1  5443  f1o2ndf1  5869  cnref1o  8733  frec2uzf1od  9408