Theorem dff13 5769

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 5421	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧))
2		ffn 5366	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		vex 2741	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
4		vex 2741	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
5	3, 4	breldm 4832	. . . . . . . . . . . . . 14 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
6		fndm 5316	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
7	6	eleq2d 2247	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
8	5, 7	imbitrid 154	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑧 → 𝑥 ∈ 𝐴))
9		vex 2741	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
10	9, 4	breldm 4832	. . . . . . . . . . . . . 14 ⊢ (𝑦𝐹𝑧 → 𝑦 ∈ dom 𝐹)
11	6	eleq2d 2247	. . . . . . . . . . . . . 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 394	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧))))
15		eqcom 2179	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑧)
16		fnbrfvb 5557	. . . . . . . . . . . . . . 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 2179	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = 𝑧)
19		fnbrfvb 5557	. . . . . . . . . . . . . . 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 606	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) ↔ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧)))
22	21	anandis 592	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) ↔ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧)))
23	22	pm5.32da 452	. . . . . . . . . . 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 1824	. . . . . . 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 1873	. . . . . . . 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 1883	. . . . . . . . . . 11 ⊢ (∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦))
31		funfvex 5533	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ V)
32	31	funfni 5317	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝐹‘𝑥) ∈ V)
33		eqvincg 2862	. . . . . . . . . . . . 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 479	. . . . . . . . 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 443	. . . . . . . 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 1867	. . . . 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 4007	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑦𝐹𝑧))
43	42	mo4 2087	. . . . . . 7 ⊢ (∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
44	43	albii 1470	. . . . . 6 ⊢ (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑧∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
45		alrot3 1485	. . . . . 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 2496	. . . . 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 454	. 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)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353  ∃wex 1492  ∃*wmo 2027   ∈ wcel 2148  ∀wral 2455  Vcvv 2738   class class class wbr 4004  dom cdm 4627   Fn wfn 5212  ⟶wf 5213  –1-1→wf1 5214  ‘cfv 5217

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fv 5225

This theorem is referenced by:  f1veqaeq  5770  dff13f  5771  dff1o6  5777  fcof1  5784  f1o2ndf1  6229  cc2lem  7265  cnref1o  9650  frec2uzf1od  10406  iseqf1olemqf1o  10493  reeff1  11708  crth  12224  eulerthlemh  12231  1arith  12365  nninfdclemf1  12453  xpsff1o  12768  ioocosf1o  14278  lgseisenlem2  14454  peano4nninf  14758  exmidsbthrlem  14773