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Theorem dff13f 5746
 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 5745 . 2
2 dff13f.2 . . . . . . . . 9
3 nfcv 2420 . . . . . . . . 9
42, 3nffv 5493 . . . . . . . 8
5 nfcv 2420 . . . . . . . . 9
62, 5nffv 5493 . . . . . . . 8
74, 6nfeq 2427 . . . . . . 7
8 nfv 1605 . . . . . . 7
97, 8nfim 1771 . . . . . 6
10 nfv 1605 . . . . . 6
11 fveq2 5486 . . . . . . . 8
1211eqeq2d 2295 . . . . . . 7
13 eqeq2 2293 . . . . . . 7
1412, 13imbi12d 311 . . . . . 6
159, 10, 14cbvral 2761 . . . . 5
1615ralbii 2568 . . . 4
17 nfcv 2420 . . . . . 6
18 dff13f.1 . . . . . . . . 9
19 nfcv 2420 . . . . . . . . 9
2018, 19nffv 5493 . . . . . . . 8
21 nfcv 2420 . . . . . . . . 9
2218, 21nffv 5493 . . . . . . . 8
2320, 22nfeq 2427 . . . . . . 7
24 nfv 1605 . . . . . . 7
2523, 24nfim 1771 . . . . . 6
2617, 25nfral 2597 . . . . 5
27 nfv 1605 . . . . 5
28 fveq2 5486 . . . . . . . 8
2928eqeq1d 2292 . . . . . . 7
30 eqeq1 2290 . . . . . . 7
3129, 30imbi12d 311 . . . . . 6
3231ralbidv 2564 . . . . 5
3326, 27, 32cbvral 2761 . . . 4
3416, 33bitri 240 . . 3
3534anbi2i 675 . 2
361, 35bitri 240 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 176   wa 358   wceq 1623  wnfc 2407  wral 2544  wf 5217  wf1 5218  cfv 5221 This theorem is referenced by:  f1mpt  5747  dom2lem  6897  dff1o6f  24502 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fv 5229
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