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Theorem dff13f 6168
 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 6167 . 2
2 dff13f.2 . . . . . . . . 9
3 nfcv 2619 . . . . . . . . 9
42, 3nffv 5879 . . . . . . . 8
5 nfcv 2619 . . . . . . . . 9
62, 5nffv 5879 . . . . . . . 8
74, 6nfeq 2630 . . . . . . 7
8 nfv 1708 . . . . . . 7
97, 8nfim 1921 . . . . . 6
10 nfv 1708 . . . . . 6
11 fveq2 5872 . . . . . . . 8
1211eqeq2d 2471 . . . . . . 7
13 equequ2 1800 . . . . . . 7
1412, 13imbi12d 320 . . . . . 6
159, 10, 14cbvral 3080 . . . . 5
1615ralbii 2888 . . . 4
17 nfcv 2619 . . . . . 6
18 dff13f.1 . . . . . . . . 9
19 nfcv 2619 . . . . . . . . 9
2018, 19nffv 5879 . . . . . . . 8
21 nfcv 2619 . . . . . . . . 9
2218, 21nffv 5879 . . . . . . . 8
2320, 22nfeq 2630 . . . . . . 7
24 nfv 1708 . . . . . . 7
2523, 24nfim 1921 . . . . . 6
2617, 25nfral 2843 . . . . 5
27 nfv 1708 . . . . 5
28 fveq2 5872 . . . . . . . 8
2928eqeq1d 2459 . . . . . . 7
30 equequ1 1799 . . . . . . 7
3129, 30imbi12d 320 . . . . . 6
3231ralbidv 2896 . . . . 5
3326, 27, 32cbvral 3080 . . . 4
3416, 33bitri 249 . . 3
3534anbi2i 694 . 2
361, 35bitri 249 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1395  wnfc 2605  wral 2807  wf 5590  wf1 5591  cfv 5594 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fv 5602 This theorem is referenced by:  f1mpt  6170  dom2lem  7574
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