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Theorem mptfnf 5709
 Description: The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)
Hypothesis
Ref Expression
mptfnf.0
Assertion
Ref Expression
mptfnf

Proof of Theorem mptfnf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eueq 3198 . . 3
21ralbii 2823 . 2
3 r19.26 2904 . . 3
4 eu5 2345 . . . 4
54ralbii 2823 . . 3
6 df-mpt 4456 . . . . . 6
76fneq1i 5680 . . . . 5
8 df-fn 5592 . . . . 5
97, 8bitri 257 . . . 4
10 moanimv 2380 . . . . . . 7
1110albii 1699 . . . . . 6
12 funopab 5622 . . . . . 6
13 df-ral 2761 . . . . . 6
1411, 12, 133bitr4ri 286 . . . . 5
15 eqcom 2478 . . . . . 6
16 dmopab 5051 . . . . . . . 8
17 19.42v 1842 . . . . . . . . 9
1817abbii 2587 . . . . . . . 8
1916, 18eqtri 2493 . . . . . . 7
2019eqeq1i 2476 . . . . . 6
21 pm4.71 642 . . . . . . . 8
2221albii 1699 . . . . . . 7
23 df-ral 2761 . . . . . . 7
24 mptfnf.0 . . . . . . . 8
2524abeq2f 2639 . . . . . . 7
2622, 23, 253bitr4i 285 . . . . . 6
2715, 20, 263bitr4ri 286 . . . . 5
2814, 27anbi12i 711 . . . 4
29 ancom 457 . . . 4
309, 28, 293bitr2i 281 . . 3
313, 5, 303bitr4ri 286 . 2
322, 31bitr4i 260 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376  wal 1450   wceq 1452  wex 1671   wcel 1904  weu 2319  wmo 2320  cab 2457  wnfc 2599  wral 2756  cvv 3031  copab 4453   cmpt 4454   cdm 4839   wfun 5583   wfn 5584 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-fun 5591  df-fn 5592 This theorem is referenced by:  fnmptf  5710
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