ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dff13f Structured version   Unicode version

Theorem dff13f 5352
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  F/_ F
dff13f.2  F/_ F
Assertion
Ref Expression
dff13f  F : -1-1->  F : -->  F `  F `
Distinct variable group:   ,,
Allowed substitution hints:   (,)    F(,)

Proof of Theorem dff13f
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 5350 . 2  F : -1-1->  F : -->  F `  F `
2 dff13f.2 . . . . . . . . 9  F/_ F
3 nfcv 2175 . . . . . . . . 9  F/_
42, 3nffv 5128 . . . . . . . 8  F/_ F `
5 nfcv 2175 . . . . . . . . 9  F/_
62, 5nffv 5128 . . . . . . . 8  F/_ F `
74, 6nfeq 2182 . . . . . . 7  F/ F `  F `
8 nfv 1418 . . . . . . 7  F/
97, 8nfim 1461 . . . . . 6  F/ F `  F `
10 nfv 1418 . . . . . 6  F/ F `  F `
11 fveq2 5121 . . . . . . . 8  F `  F `
1211eqeq2d 2048 . . . . . . 7  F `  F `  F `  F `
13 equequ2 1596 . . . . . . 7
1412, 13imbi12d 223 . . . . . 6  F `  F `  F `  F `
159, 10, 14cbvral 2523 . . . . 5  F `  F `  F `
 F `
1615ralbii 2324 . . . 4  F `
 F `  F `
 F `
17 nfcv 2175 . . . . . 6  F/_
18 dff13f.1 . . . . . . . . 9  F/_ F
19 nfcv 2175 . . . . . . . . 9  F/_
2018, 19nffv 5128 . . . . . . . 8  F/_ F `
21 nfcv 2175 . . . . . . . . 9  F/_
2218, 21nffv 5128 . . . . . . . 8  F/_ F `
2320, 22nfeq 2182 . . . . . . 7  F/ F `  F `
24 nfv 1418 . . . . . . 7  F/
2523, 24nfim 1461 . . . . . 6  F/ F `  F `
2617, 25nfralxy 2354 . . . . 5  F/  F `  F `
27 nfv 1418 . . . . 5  F/  F `  F `
28 fveq2 5121 . . . . . . . 8  F `  F `
2928eqeq1d 2045 . . . . . . 7  F `  F `  F `  F `
30 equequ1 1595 . . . . . . 7
3129, 30imbi12d 223 . . . . . 6  F `  F `  F `  F `
3231ralbidv 2320 . . . . 5  F `  F `  F `
 F `
3326, 27, 32cbvral 2523 . . . 4  F `
 F `  F `
 F `
3416, 33bitri 173 . . 3  F `
 F `  F `
 F `
3534anbi2i 430 . 2  F : -->  F `  F `  F : -->  F `  F `
361, 35bitri 173 1  F : -1-1->  F : -->  F `  F `
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   F/_wnfc 2162  wral 2300   -->wf 4841   -1-1->wf1 4842   ` cfv 4845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fv 4853
This theorem is referenced by:  f1mpt  5353  dom2lem  6188
  Copyright terms: Public domain W3C validator