Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dff13f Unicode version

Theorem dff13f 5965
 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
dff13f.2
Assertion
Ref Expression
dff13f
Distinct variable group:   ,,
Allowed substitution hints:   (,)   (,)

Proof of Theorem dff13f
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dff13 5963 . 2
2 dff13f.2 . . . . . . . . 9
3 nfcv 2540 . . . . . . . . 9
42, 3nffv 5694 . . . . . . . 8
5 nfcv 2540 . . . . . . . . 9
62, 5nffv 5694 . . . . . . . 8
74, 6nfeq 2547 . . . . . . 7
8 nfv 1626 . . . . . . 7
97, 8nfim 1828 . . . . . 6
10 nfv 1626 . . . . . 6
11 fveq2 5687 . . . . . . . 8
1211eqeq2d 2415 . . . . . . 7
13 equequ2 1694 . . . . . . 7
1412, 13imbi12d 312 . . . . . 6
159, 10, 14cbvral 2888 . . . . 5
1615ralbii 2690 . . . 4
17 nfcv 2540 . . . . . 6
18 dff13f.1 . . . . . . . . 9
19 nfcv 2540 . . . . . . . . 9
2018, 19nffv 5694 . . . . . . . 8
21 nfcv 2540 . . . . . . . . 9
2218, 21nffv 5694 . . . . . . . 8
2320, 22nfeq 2547 . . . . . . 7
24 nfv 1626 . . . . . . 7
2523, 24nfim 1828 . . . . . 6
2617, 25nfral 2719 . . . . 5
27 nfv 1626 . . . . 5
28 fveq2 5687 . . . . . . . 8
2928eqeq1d 2412 . . . . . . 7
30 equequ1 1692 . . . . . . 7
3129, 30imbi12d 312 . . . . . 6
3231ralbidv 2686 . . . . 5
3326, 27, 32cbvral 2888 . . . 4
3416, 33bitri 241 . . 3
3534anbi2i 676 . 2
361, 35bitri 241 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1649  wnfc 2527  wral 2666  wf 5409  wf1 5410  cfv 5413 This theorem is referenced by:  f1mpt  5966  dom2lem  7106 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fv 5421
 Copyright terms: Public domain W3C validator