Theorem dff13 5974

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 5597	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧))
2		ffn 5533	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		vex 2824	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
4		vex 2824	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
5	3, 4	breldm 4985	. . . . . . . . . . . . . 14 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
6		fndm 5480	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
7	6	eleq2d 2308	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
8	5, 7	imbitrid 154	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑧 → 𝑥 ∈ 𝐴))
9		vex 2824	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
10	9, 4	breldm 4985	. . . . . . . . . . . . . 14 ⊢ (𝑦𝐹𝑧 → 𝑦 ∈ dom 𝐹)
11	6	eleq2d 2308	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
12	10, 11	imbitrid 154	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑦𝐹𝑧 → 𝑦 ∈ 𝐴))
13	8, 12	anim12d 335	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
14	13	pm4.71rd 398	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧))))
15		eqcom 2240	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑧)
16		fnbrfvb 5741	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑧 ↔ 𝑥𝐹𝑧))
17	15, 16	bitrid 192	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑧 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑧))
18		eqcom 2240	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = 𝑧)
19		fnbrfvb 5741	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) = 𝑧 ↔ 𝑦𝐹𝑧))
20	18, 19	bitrid 192	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧 = (𝐹‘𝑦) ↔ 𝑦𝐹𝑧))
21	17, 20	bi2anan9 614	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) ↔ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧)))
22	21	anandis 600	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) ↔ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧)))
23	22	pm5.32da 456	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧))))
24	14, 23	bitr4d 191	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)))))
25	24	imbi1d 231	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))) → 𝑥 = 𝑦)))
26		impexp 263	. . . . . . . . 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	bitrdi 196	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦))))
28	27	albidv 1877	. . . . . . 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 1926	. . . . . . . 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		19.23v 1936	. . . . . . . . . . 11 ⊢ (∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦))
31		funfvex 5712	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
32	31	funfni 5483	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
33		eqvincg 2950	. . . . . . . . . . . . 13 ⊢ ((𝐹‘𝑥) ∈ V → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))))
34	32, 33	syl 14	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))))
35	34	imbi1d 231	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦)))
36	30, 35	bitr4id 199	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
37	36	adantrr 483	. . . . . . . . 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 447	. . . . . . . 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	bitrid 192	. . . . . . 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 188	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
41	40	2albidv 1920	. . . . 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 4133	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑦𝐹𝑧))
43	42	mo4 2148	. . . . . . 7 ⊢ (∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
44	43	albii 1523	. . . . . 6 ⊢ (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑧∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
45		alrot3 1538	. . . . . 6 ⊢ (∀𝑧∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
46	44, 45	bitri 184	. . . . 5 ⊢ (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
47		r2al 2569	. . . . 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 223	. . . 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 458	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
51	1, 50	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  ∀wal 1400   = wceq 1402  ∃wex 1545  ∃*wmo 2087   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   class class class wbr 4130  dom cdm 4774   Fn wfn 5372  ⟶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:  f1veqaeq  5975  dff13f  5976  dff1o6  5982  fcof1  5989  f1o2ndf1  6464  fdcf1  7316  cc2lem  7633  cnref1o  10062  frec2uzf1od  10858  iseqf1olemqf1o  10958  reeff1  12486  crth  13025  eulerthlemh  13032  1arith  13169  nninfdclemf1  13395  xpsff1o  13723  ghmf1  14129  kerf1ghm  14130  znf1o  15070  ioocosf1o  16047  mpodvdsmulf1o  16245  gausslemma2dlem1f1o  16345  lgseisenlem2  16356  2lgslem1b  16374  peano4nninf  17215  exmidsbthrlem  17233