Theorem dff13 7209

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 6735	. 2 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧))
2		ffn 6668	. . . 4 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴)
3		vex 3433	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
4		vex 3433	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
5	3, 4	breldm 5863	. . . . . . . . . . . . . 14 ⊢ (𝑥𝐹𝑧 → 𝑥 ∈ dom 𝐹)
6		fndm 6601	. . . . . . . . . . . . . . 15 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
7	6	eleq2d 2822	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ 𝐴))
8	5, 7	imbitrid 244	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑥𝐹𝑧 → 𝑥 ∈ 𝐴))
9		vex 3433	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
10	9, 4	breldm 5863	. . . . . . . . . . . . . 14 ⊢ (𝑦𝐹𝑧 → 𝑦 ∈ dom 𝐹)
11	6	eleq2d 2822	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ dom 𝐹 ↔ 𝑦 ∈ 𝐴))
12	10, 11	imbitrid 244	. . . . . . . . . . . . 13 ⊢ (𝐹 Fn 𝐴 → (𝑦𝐹𝑧 → 𝑦 ∈ 𝐴))
13	8, 12	anim12d 610	. . . . . . . . . . . 12 ⊢ (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)))
14	13	pm4.71rd 562	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧))))
15		eqcom 2743	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑥) ↔ (𝐹‘𝑥) = 𝑧)
16		fnbrfvb 6890	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝐹‘𝑥) = 𝑧 ↔ 𝑥𝐹𝑧))
17	15, 16	bitrid 283	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑧 = (𝐹‘𝑥) ↔ 𝑥𝐹𝑧))
18		eqcom 2743	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝐹‘𝑦) ↔ (𝐹‘𝑦) = 𝑧)
19		fnbrfvb 6890	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) = 𝑧 ↔ 𝑦𝐹𝑧))
20	18, 19	bitrid 283	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑧 = (𝐹‘𝑦) ↔ 𝑦𝐹𝑧))
21	17, 20	bi2anan9 639	. . . . . . . . . . . . 13 ⊢ (((𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) ↔ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧)))
22	21	anandis 679	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) ↔ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧)))
23	22	pm5.32da 579	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧))))
24	14, 23	bitr4d 282	. . . . . . . . . 10 ⊢ (𝐹 Fn 𝐴 → ((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)))))
25	24	imbi1d 341	. . . . . . . . 9 ⊢ (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦))) → 𝑥 = 𝑦)))
26		impexp 450	. . . . . . . . 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 287	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → (((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦))))
28	27	albidv 1922	. . . . . . 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 1941	. . . . . . . 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 1944	. . . . . . . . . 10 ⊢ (∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦))
31		fvex 6853	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑥) ∈ V
32	31	eqvinc 3591	. . . . . . . . . . 11 ⊢ ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ ∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)))
33	32	imbi1i 349	. . . . . . . . . 10 ⊢ (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ (∃𝑧(𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦))
34	30, 33	bitr4i 278	. . . . . . . . 9 ⊢ (∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦) ↔ ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))
35	34	imbi2i 336	. . . . . . . 8 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ∀𝑧((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
36	29, 35	bitri 275	. . . . . . 7 ⊢ (∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑧 = (𝐹‘𝑥) ∧ 𝑧 = (𝐹‘𝑦)) → 𝑥 = 𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
37	28, 36	bitrdi 287	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → (∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
38	37	2albidv 1925	. . . . 5 ⊢ (𝐹 Fn 𝐴 → (∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦))))
39		breq1 5088	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥𝐹𝑧 ↔ 𝑦𝐹𝑧))
40	39	mo4 2566	. . . . . . 7 ⊢ (∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
41	40	albii 1821	. . . . . 6 ⊢ (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑧∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
42		alrot3 2166	. . . . . 6 ⊢ (∀𝑧∀𝑥∀𝑦((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
43	41, 42	bitri 275	. . . . 5 ⊢ (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥∀𝑦∀𝑧((𝑥𝐹𝑧 ∧ 𝑦𝐹𝑧) → 𝑥 = 𝑦))
44		r2al 3173	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
45	38, 43, 44	3bitr4g 314	. . . 4 ⊢ (𝐹 Fn 𝐴 → (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
46	2, 45	syl 17	. . 3 ⊢ (𝐹:𝐴⟶𝐵 → (∀𝑧∃*𝑥 𝑥𝐹𝑧 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
47	46	pm5.32i 574	. 2 ⊢ ((𝐹:𝐴⟶𝐵 ∧ ∀𝑧∃*𝑥 𝑥𝐹𝑧) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
48	1, 47	bitri 275	1 ⊢ (𝐹:𝐴–1-1→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃*wmo 2537  ∀wral 3051   class class class wbr 5085  dom cdm 5631   Fn wfn 6493  ⟶wf 6494  –1-1→wf1 6495  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fv 6506

This theorem is referenced by:  dff13f  7210  f1veqaeq  7211  fpropnf1  7222  dff14a  7225  dff1o6  7230  fcof1  7242  nf1const  7259  soisoi  7283  f1opr  7423  f1o2ndf1  8072  fnwelem  8081  smo11  8304  tz7.48lem  8380  omsmo  8594  unxpdomlem3  9168  unfilem2  9216  fofinf1o  9242  inf3lem6  9554  r111  9699  fseqenlem1  9946  fodomacn  9978  alephf1  10007  alephiso  10020  ackbij1lem17  10157  infpssrlem5  10229  fin23lem28  10262  fin1a2lem2  10323  fin1a2lem4  10325  axcc2lem  10358  domtriomlem  10364  cnref1o  12935  injresinj  13746  om2uzf1oi  13915  cshf1  14772  wwlktovf1  14919  reeff1  16087  bitsf1  16415  crth  16748  eulerthlem2  16752  1arith  16898  vdwlem12  16963  xpsff1o  17531  setcmon  18054  fthestrcsetc  18116  embedsetcestrclem  18123  fthsetcestrc  18131  yoniso  18251  chnpof1  18596  ghmf1  19221  kerf1ghm  19222  orbsta  19288  symgextf1  19396  symgfixf1  19412  odf1  19537  znf1o  21531  cygznlem3  21549  uvcf1  21772  lindff1  21800  mvrf1  21964  ply1sclf1  22254  scmatf1  22496  mdetunilem8  22584  mat2pmatf1  22694  pm2mpf1  22764  ist0-4  23694  ovolicc2lem4  25487  recosf1o  26499  efif1olem4  26509  basellem4  27047  mpodvdsmulf1o  27157  dvdsmulf1o  27159  lgsqrlem2  27310  lgseisenlem2  27339  2lgslem1b  27355  negsf1o  28046  oniso  28263  om2noseqf1o  28293  bdayn0sf1o  28362  axlowdimlem15  29025  upgrwlkdvdelem  29804  wlkswwlksf1o  29947  wwlksnextinj  29967  clwlkclwwlkf1  30080  clwwlkf1  30119  frgrncvvdeqlem8  30376  numclwwlk1lem2f1  30427  pjmf1  31787  unopf1o  31987  2ndresdju  32722  fnpreimac  32743  s3f1  33007  ccatf1  33009  swrdf1  33016  mndlactf1  33086  mndractf1  33088  tocyccntz  33205  extvfvcl  33680  dff15  35227  f1resrcmplf1d  35230  f1resfz0f1d  35296  erdszelem9  35381  mrsubff1  35696  msubff1  35738  mvhf1  35741  f1omptsnlem  37652  fvineqsnf1  37726  fvineqsneu  37727  poimirlem26  37967  poimirlem27  37968  grpokerinj  38214  cdleme50f1  40989  dihf11  41713  hashscontpow  42561  hashnexinj  42567  aks6d1c5  42578  sticksstones2  42586  aks6d1c6lem3  42611  fimgmcyc  42979  dnnumch3  43475  wessf1ornlem  45615  projf1o  45626  sumnnodd  46060  dvnprodlem1  46374  fourierdlem34  46569  fourierdlem51  46585  fsetsnf1  47500  cfsetsnfsetf1  47507  fcoresf1  47517  imasetpreimafvbijlemf1  47864  fargshiftf1  47901  sprsymrelf1  47956  prproropf1o  47967  fmtnof1  47998  prmdvdsfmtnof1  48050  uspgrsprf1  48623  1arymaptf1  49118  2arymaptf1  49129  rrx2xpref1o  49194  oppff1  49623  diag1f1  49782  diag2f1  49784